Testable scenario for relativity with minimum length

I propose a general class of spacetimes whose structure is governed by observer-independent scales of both velocity (c) and length (Planck length), and I observe that these spacetimes can naturally host a modification of FitzGerald–Lorentz contraction such that lengths which in their inertial rest frame are bigger than a “minimum length” are also bigger than the minimum length in all other inertial frames. With an analysis in leading order in the minimum length, I show that this is the case in a specific illustrative example of postulates for relativity with velocity and length observer-independent scales.