As @Luke says, they will always be at relative rest.... but only in the LAB frame.
The distinction is necessary because the two simultaneous events that are used for comparison are spatially separated, and thus depend on the frame of reference.
In the problem stated, the worldlines of each traveler are congruent hyperbolas (corresponding to the same proper-acceleration).
In the LAB frame, the situation can be described as one worldline is a translation in space of the other.
So, at any time T in the LAB frame,
the two simultaneous events on the traveller worldlines have parallel tangent lines..
and this distance between corresponding events doesn't change with T.
But in the frame of the LEFT-TRAVELER (ALICE),
those LAB-simultaneous events aren't simultaneous.
Instead, in the LEFT-TRAVELER (ALICE) uses a different pair of simultaneous events on the traveler worldlines... and their tangent lines are not parallel.... they don't have the same velocity in this frame.