Since
@Ibix insists on conventionality of synchronization, we can also allow ourselves some free – thinking. At least, the astronauts can ascribe themselves state of proper motion instead of proper rest.
Everything becomes … a bit simpler as soon as the astronauts in the spaceships admit their own motion in the frame S and don't jump from frame to frame like fleas. In this case they can explain scattering of the spaceships by means of contraction of their own measurement ruler.
All spaceships – A, B, C are initially at rest. When the spaceships B and C start moving, their rulers contract, and since they measure distance between spaceships with Lorentz - contracted ruler, they make conclusion that the spaceships scatter from each other. Obviously, they cannot detect contraction of the ruler, since they contract themselves. They simply see, that the more they accelerate, the more rulers they can fit between spaceships. Thread also contracts and breaks.
It is interesting to consider second stage of acceleration, when the spaceships turn their engines at 180 degrees with the aim to get back into the previous position in S.
According to SR, their clocks will be out of sync (in S’) after first cycle of acceleration. What time their clocks will show? Their clocks will show a bit less time (due time dilation), that in S frame, but they will keep S – simultaneity anyway. For example, S - time is 12PM and the both clocks show 11AM.
But, since they move in S, velocity of light towards B and C from the middle point in will be different and if spaceships will simultaneously ( by S time) will launch flashes, the beams will come to middle point at different moments. So, an "Einsteinian" observer in the middle point (between B and C) will conclude, that flashes were launched at different moments and clocks are out of sync.
If they wish to repeat the experiment with the same initial (Bell’s) conditions, they have to re-synchronize their clocks (to start engines simultaneously in S’) by Einstein technique. But, in this case they will scatter even further instead of coming into initial position. They will scatter further and further at every turn, if they start their engines simultaneously by Einstein.
To avoid scattering when they go back, they have to use “conventionally uniform time of reference frame S ” and start engines simultaneously by S – time, or simply leave clocks alone and not to re-synchronize.
For the sake of convenience, we can stay in S frame. In this frame their clocks will actually show the same time after acceleration. If they want to return back to their original position, they simple reverse engines and start them simultaneously according to their clock readings, or at the same moment by S time.
But, they can “forget” their previous history. They may think, that since they cannot detect their own motion (detect anisotropy of light) so, they can re-synchronize clocks by Einstein, what they finally do.
The spaceship A (like in Bell’s arrangement) in the middle between them sends beams and they start engines “simultaneously” in their frame. However, since they now in motion in S frame, the beam will reach each of them at different moments of S - time. The “back” ship earlier and the “front” ship later. So, their “S’ simultaneity” is no longer “simultaneity” in S frame. They will start engines at different moments and scatter again. Thread breaks.
After the turn the spaceships will actually scatter due to Einsteinian simultaneity in S’, which is not simultaneity in S at all (engines start at different moments of S-time), though their length comes back to L and ruler increases back accordingly.