I want to derive the distance between spaceships in Bell's paradox as a function of proper time of the trailing spaceship, in the case of constant proper acceleration of both space ships.

My attempt didn't work, as it gave a distance that gets shorter with time. I would like to know why it didn't work, and what the actual solution is.

My attempt:

First, define the trajectory of the trailing spaceship as a hyperbolic trajectory with a rindler coordinate frame. In the inertial frame: $x\_T = \frac1\alpha \cosh(\alpha \bar t)$

In the inertial frame, the leading spaceship has the same hyperbolic trajectory except it is spatially shifted to the right by the length of the rope $l$. Hence, the difference in the spatial coordinates is always $x\_L - x\_T = l$ as measured in the inertial frame.
Next, use the Rindler coordinate transformation to express the difference in coordinates in terms of $\bar x$, given that the time of measurement is $\bar t$ .
$$x = \bar x \cosh(\alpha \bar t) \implies x\_L - x\_T = (\bar x\_L - \bar x\_T)\cosh(\alpha \bar t) \implies \bar x\_L - \bar x\_T = \frac l{\cosh(\alpha \bar t)}$$

This expression says that the distance between the spaceships in the trailing spaceship's MCRF is decreasing with time, meaning the rope will not break which is wrong. What is the mistake?

Thank you