The dismantling characterization of trivial matroid homomorphism reconfiguration

Let $N$ be a matroid, and let $\operatorname{Recol}_{\mathbb{M}}(N)$ denote the reconfiguration problem for matroid homomorphisms into $N$. A matroid dismantles to a retract $N'$ if a series of dismantling retractions has composition taking $N$ to $N'$, where a dismantling retraction is a retraction adjacent to the identity map. Write $M^\ell(K_1)$ for the loop on one vertex and $M(K_2)$ for the matroid of an edge.

Dismantling characterization. $\operatorname{Recol}_{\mathbb{M}}(N)$ is trivial if and only if $N$ dismantles to the loop $M^\ell(K_1)$ or the edge $M(K_2)$.

This would extend the corresponding characterization for graphs, where triviality of the reconfiguration problem is equivalent to dismantlability. The preceding results establish that dismantling to either of these two small matroids is sufficient; the converse is posed as an analogue for matroids and remains open.