 Solution

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 $x=a+L$.