::::Listen, S' is the new rest system of the rockets after the acceleration is finished. Why is this so hard to understand? So if you perform a Lorentz transformation of points A and B, from system S (where the ships are moving with v after acceleration) to S' (where they are at rest after acceleration), you are calculating the endpoints of resting objects in S', thus you have the rest length of the objects and the distance between the endpoints. And what do the Lorentz-Trafos tell us? Well they unambiguously show that the distance between A and B (again, those are the coordinates of the endpoints of in S' resting objects) has increased as compared to the coordinate length between A and B in S. Of course, this is in perfect agreement with relativity of simultaneity, which shows that the spaceships started at different times in S', thus they had different accelerations... 
 ::::Summarizing: 1) You have two endpoints A and B with different accelerations in S', therefore they are moving away from each other in S', and after the acceleration the distance of the now resting objects has increased, and the string has been stretched. 2) Of course, one can also choose frame S in which the accelerations "appeared" to be the same due to relativity of simultaneity, and therefore the increased distance in S' "apparently" stays the same in S. (Actually, one should abandon the word "appear" in relativity, it serves no purpose and only distracts one from correctly applying the Lorentztransformation.)  
 ::::Now, Dewan, Beran, and Bell unfortunately started with 2) and only afterwards at 1), so they had to focus their attention mostly on S and on length contraction, which caused much confusion by some readers. But the paradox is much easier to understand if one (like Petkov and Franklin) starts with 1) and then look at 2). 
 ::::BTW: As explained above, modern authors understand the paradox very well, and some of them even try to confirm it experimentally ("Bell's spaceship paradox in electron interferometry", ).