Duality invariance of matroid p-characteristic sets

Let $M$ be a matroid and let $M^*$ be its dual. Write $4\chi_M4$ for the p-characteristic set of $M$.

Duality conjecture.

$$\chi_{M^*}=\chi_M.$$

This would extend the equality already known for prime-power characteristics, where representability of a matroid over a finite field is equivalent to representability of its dual over the same field. The conjecture asks whether the corresponding equality holds for all integers $v\ge 2$; the construction of a $4\mathrm{VOA}(M^*,v)4$ from a $4\mathrm{VOA}(M,v)4$ is not established in the supplied text.