Polynomial hitting-number conjecture for complete regular matroids

Let $M$ be a complete regular matroid of rank $r$. Here, complete means that a totally unimodular representation of $M$ is maximally totally unimodular: adding any column that is not a multiple of an existing column violates total unimodularity. Let $h(M)$ denote the hitting number of $M$.

Polynomial hitting-number conjecture. The quantity $h(M)$ is bounded by a polynomial in $r$.

This conjecture asks for a polynomial bound on the hitting number for complete regular matroids, bypassing the decomposition of regular matroids into $1$-, $2$-, and $3$-sums. The source does not state whether the conjecture is known or open; in particular, the analogous bound for $3$-sums is described as unclear.