I analysed this in some detail in this old thread (with yet more diagrams) https://www.physicsforums.com/showthread.php?t=216113 and concluded that two clocks would be out of sync after born rigid accleration when they reach same final constant velocity.Hi Kev. I think you have these two parts slightly wrong. The Born-accelerated observers on your 'floors' do not agree that their respective clock rates are the same, because their clocks get more and more out of sync during the acceleration. Think what will happen in your Rindler diagram when the acceleration is stopped - don't you think the clocks will then need to be resynchronized?
In this thread I am considering the slightly different situation where the two clocks maintain constant born rigid acceleration indefinitely. I would be interested in your opinion of my conclusion in that old thread ;)
The light signals sent forward and backward should behave as any other light signal in a time space diagram moving at 45 degrees as I have shown on this diagram (the blue lines). If the signals are sent at 0 seconds by the upper and and lower observers and both arrive at 1 seconds as shown by their respective clock and then return at 2 seconds as shown by their respective clocks then presumably the upper and lower clocks must remain in sync (while constant proper born rigid acceleration is maintained) as far as I can tell.Your purple lines of "instantaneous simultaneity" is when the clocks on all floors read the same time, but they realize that the clocks get out of sync. Light signals send forward and backward during the acceleration are red-shifted and blue-shifted respectively, just like in a gravitational field. It's only the tidal effects that differ, AFAIK.