Then what you're talking about has nothing to do with gravitational time dilation or the equivalence principle. Gravitational time dilation is about comparing clocks at different locations within a gravitational field. Muon decay works like a clock, it will be observed to happen slower the lower the muon is in a gravitational field.

The equivalence principle says the same thing about accelerating rockets: muons at the rear of a rocket decay slower than muons at the front of the rocket.

You're asking questions, but you aren't accepting the answers. You are misunderstanding time dilation and muon experiments.

What relativity predicts is that muons decay slower the lower in a gravitational field than they do higher in a gravitational field, and it predicts that the same is true of muons in the front/rear of rockets. That's what the equivalence principle says.

No offense, but you're asking questions that you don't have the background to understand. People have tried to give you the information, but you have rejected it. So that makes it frustrating.

I want to make a correction. I don't think that any experiments have shown the effect of gravitational time dilation on muon decay. What has been shown is just regular velocity-dependent time dilation, and the gravity of the Earth has a negligible effect. So I think that the basis for your original question is not well-founded.

Good point. Experiments with muons showed only velocity dependent time dilation and absence of time dilation in acceleration. Time dilation effect in gravity of the Earth would be negligible and it was only my conjecture that there should be also time dilation effect.

The big problem is where you state that you will ignore the front and rear potential time difference in the spaceship. You can't make a comparison to gravitational time dilation at the Earth's surface without taking into a account this potential time difference, Because gravitational time dilation is a measurement of clocks at different potentials. To make any comparison, you would have to do the following type of experiment:

At the Earth, you have two identical clocks. One sitting on the surface and one 10 meters above it. On the ship, you have two identical clocks, one at the rear and one 10 meters closer to the nose. You then compare how much the clocks on Earth differ after the lower clock has ticked off some interval to how much the clocks on the ship differ after the rear clock ticks off the same interval.

This difference will not be the same, it will turn out that the clocks on the ship will differ from each other just a tad more than those on the Earth. This is because the potential between the clocks is the equivalent of that for a gravity field that maintains a constant 1g over the distance separating the clocks, while on the Earth, gravity falls off slightly between the two heights of the clocks. Thus you get a larger potential difference in the ship than you do for the Earth.

This does not violate the Equivalence principle however. For example, what if I have another two clock, again 10 meters apart, but this time the lower clock is twice the radius of the Earth away from a planet with 4 times the Earth's mass. That lower clock would be at 1g of gravity just like the Earth surface clock is. However, since 10 meters is a smaller fraction of 2 Earth radii than 1 Earth radius, gravity will fall off less between these two clocks, and the potential between them will be larger than that for the Earth clocks, and thus the difference in tick rate between this pair will be larger than for the Earth pair. In both cases the differential tick rate is caused by gravitational time dilation, and even though the lower clock of each pair is experiencing the same force of gravity, the differential between of the clocks of each pair is different.

You will also notice that our new pair of clocks more closely matches the clocks in the spaceship than the Earth clocks do. If we increase their distance from the planet while simultaneously increasing the mass of the planet in order to maintain 1g at the lower clock, the behavior of these clocks become closer and closer to that of those in the ship. The behavior of these two set of clocks begin to converge. At some point they will become all but indistinguishable. This is the main point of the equivalence principle: that over a small enough region gravity and acceleration are equivalent.

The muon experiments show that acceleration itself has no effect on clock rates. But all of the muons in those experiments (the ones where the muons were trapped in a storage ring) were at the same "height" relative to the acceleration, so these experiments say nothing about the rates of clocks at different "heights" in an accelerating rocket. Other experiments, such as the Pound-Rebka experiment, clearly show that clocks at different heights on Earth run at different rates, even though they are at rest relative to each other; and by the equivalence principle, the same will be true of clocks at different heights in a rocket accelerating at 1 g in free space. This difference in clock rates is not due to a difference in accelerations; it's due to a difference in height.

Yes, Im writing specifically about zero height difference and pure acceleration effect and everybody seems like repeating the only mantra they know - the difference in height in the rocket. Seems like common obsession of all who had studied physics. It surely has some hidden Freudian meaning :)

Well, you were asking about gravity's effect on time dilation. It has no measurable effect unless you compare time rates at different heights. So the reason people kept bringing that up was because you seemed to think that gravity was involved somehow.

It's very annoying--you keep asking about gravity's effect on time dilation, and people keep telling you, and you keep saying: I don't mean that. Well, there is no other effect.

Say muon lifetime is 2.2 micro second, both the Earth lab and the 1g accelerating rocket observe the same value using their clocks.
Time dilation works when we compare two positions in gravitational field, e.g. surface vs center of the Earth, upward vs downward position of the accelerating rocket.
For an example you should provide more information, height or depth of the Earth observed in the rocket system and also its speed, for time comparison of the two.

Nevertheless, we can analyze what SR/GR would predict. It predicts that they would be equal.

It means that experts naturally assume that if you are asking about gravitational time dilation then you would want to discuss a scenario where there would be measurable gravitational time dilation. Your proposed experiment is not sensitive to the acceleration/gravity.

Zero height difference in the rocket is the equivalent of zero height distance on the Earth. Zero height difference in a gravitational field produces zero time dilation. Neither gravity or acceleration produce time dilation in of themselves. Time dilation in both cases is only caused by a difference in potential. Either different altitudes in the Earth's gravity or different positions in the accelerating frame of the rocket.
When someone says that the time dilation at the Earth's surface has some value, they mean they mean compared to a point that is at a maximum gravitational potential relative to the Earth. (By convention we set this maximum as being zero, with all potentials closer to the Earth having negative values. Thus the specific gravitational potential at the Earth surface is ~ -62511759 j/kg

And we're telling you, repeatedly, that there is no such thing. So your question is answered. And there is no point in continuing to go around in circles about it.