Bounded-Jacobian-exponent conjecture for connected regular matroids

Let $M$ be a connected regular matroid, and let $\mathop{\mathrm{Jac}}\nolimits\left( M \right)$ denote its Jacobian, a finite abelian group. The exponent of this group is the smallest positive integer $m$ such that $ma=0$ for every $a\in\mathop{\mathrm{Jac}}\nolimits\left( M \right)$.

Regular-matroid Jacobian exponent conjecture. For every positive integer $k$, there are only finitely many connected regular matroids $M$ such that the exponent of $\mathop{\mathrm{Jac}}\nolimits\left( M \right)$ is at most $k$.

This is the regular-matroid analogue of the conjecture for biconnected graphs. The source states the analogous conjecture but does not give a resolution of it; the paper instead proves the exponent-$2$ case and characterizes all such matroids.