I don't know what you mean by "theorem 3"--what page are you looking at? In any case, looking over the paper, in equation 3.12 at the bottom of p. 11 they explicitly show that the length of the string does not
change in the frame where the ships have identical coordinate accelerations (and started accelerating simultaneously). I'm sure you won't find any physicists who dispute this very obvious and trivial point.
This paper does not
dispute Bell's claim that the cable would break! Instead it calls for a rethinking of the reason
the cable breaks...the author's argument seems to be that the only physical way of defining an object's length is by looking at its own rest frame, so that treating "length contraction" as a change in length is overly confusing...from p. 3:
The author then points out that the "rest frame length" of the cable or string does
grow as the ships accelerate, even though the distance between them in the observer's frame does not change, and that this should be seen as the true reason a cable or string would break, not length contraction: