How does time dilation appear to a third observer?

[SuperfundMatt]

Let’s assume it is 5,000,000 years in the future and we have created a quantum synchronization box that can broadcast a signal, used to synchronize two clocks and video feeds, no matter where they are in the universe.

The third observer watches a split TV screen, which features the faces of two clocks that have cameras affixed which broadcast the faces of back to the TV screen of the third observer, with one clock traveling at a Lorentz factor of γ = 100, and the other on Earth.

The clock traveling at γ = 100, to the third observer, should have hands that appeared to have stopped, because of the effects of time dilation, while the clock on Earth would appear to be normally functioning. Yet, the box is transmitting signal to both clocks, at the same time, keeping them synchronized.

Considering the addition of the instantaneous external synchronization, what does the third observer see, on the split TV screen, from their frame of reference, which also contains the quantum synchronization box?

 

[Nugatory]

Assuming your magic box is equivalent to assuming that there is a way of sending a signal faster than light (how else could the magic box instantaneously synchronize a distant clock?) and no such thing can exist. Thus, your question can be paraphrased as "what do the laws of physics say would happen in a situation in which the laws of physics do not apply?" and there's no meaningful answer to that question.

Variations of this question have been asked many times here in the past. This thread is closed, as there is not much more to say here.