Totally unimodular matrix flow conjecture

Determine whether the following holds: for any totally unimodular k × m matrix A, if there exists a nonzero solution x = (x1,…,xm) to Ax = 0 with at most n distinct values among its entries, then there also exists a solution x′ = (x′1,…,x′m) to Ax = 0 such that |x′i| ∈ {1,2,…,n} for all i ∈ {1,…,m}.

Background

This conjecture, posed in the context of nowhere-zero flows and regular matroids, can be seen as a weak version of the Lonely Runner Conjecture. It implies an important flow theorem (Theorem 9) when A is the vertex–edge incidence matrix of a graph.

The statement links the existence of a solution with bounded value-complexity to one with bounded integer magnitudes, leveraging the structure of totally unimodular matrices.