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Maintaining a fixed separation in one inertial laboratory frame is not the same as maintaining a fixed separation measured locally with the accelerating rockets. The attached string is prevented from undergoing the laboratory contraction that an unstrained comoving string would have. It is therefore stretched and develops tensile stress: this is Bell's spaceship paradox, not a failure of Lorentz contraction.

For sufficiently prolonged acceleration and a finite elastic breaking strain, the string breaks. If an acceleration history produces only a small bounded increase in speed, whether it actually breaks depends on its material strength; the kinematic conclusion is stretching, not an unconditional assertion that every arbitrarily weak acceleration must snap every string. Stress also propagates through a real string at a finite speed. The construction enforcing Born rigidity in the following parts is a different set of trajectories chosen to keep the comoving separation fixed.

Take $a>0$ as a length. For the rear hyperbola,The implicit front event from iii has $t_B=(1+L/a)t$ and $x_B=(1+L/a)f(t)$. Eliminate the rear time $t$ to obtainThe rear proper acceleration is $c^2/a$, while the front proper acceleration is $c^2/(a+L)$. Thus Born rigidity in the acceleration requires the front rocket to accelerate less strongly. Their constant separation $L$ is measured on the tilted instantaneous-rest slices; their equal-laboratory-time coordinate separation decreases from $L$ rather than staying constant. At $t=0$ both rockets are at rest, at $x=a$ and $x=a+L$.