Polynomiality criterion for semi-invariants in type C

Let $\operatorname{\mathfrak{q}}^C$ be a standard parabolic contraction in type $C$, and let $\operatorname{Sy}(\operatorname{\mathfrak{q}}^C)$ denote its algebra of semi-invariants. Let $\mathbf{M}_2$ be the set associated with the contraction. Polynomiality criterion conjecture. The algebra $\operatorname{Sy}(\operatorname{\mathfrak{q}}^C)$ is not polynomial if and only if there exists an odd $m\in\mathbf{M}_2$ such that $2m-2\in\mathbf{M}_2$.

This gives a necessary and sufficient condition for non-polynomiality in type $C$. It is motivated by the counterexample and the computations described in the paper; the supplied text gives no proof or resolution.