Homotopy-equivalent topological representations detect isomorphisms of matroids

Let $X$ be a finite CW complex that is not contractible, and let $(M,l)$ and $(N,l')$ be $\rho$-immersed matroids such that

$${ \mathscr T}_X(M,l) \simeq {\mathscr T}_X(N,l').$$

Suppose there exists a surjective weak map $\tau:M\to N$. Topological representation isomorphism conjecture. Then $\tau^{\#}$ is an isomorphism.

This conjecture proposes that, for a noncontractible base complex, homotopy equivalence of the associated topological representations forces the induced map on the relevant matroid data to be an isomorphism. The surrounding results establish strict Betti-number inequalities under rank-decreasing surjective weak maps, but the stated implication is not resolved here.