Factorization conjecture for semi-invariants in type C

Let $\mathbf{M}_2$ and $r'_m$ be the sets and integers associated with the standard parabolic contraction in type $C$, and let $\operatorname{pr}(F_j^\bullet)$ denote the projected semi-invariants introduced in the paper. For odd $m\in\mathbf{M}_2$, define

$$f=\begin{cases} \operatorname{pr}(F_{2m-2}^\bullet)-\frac{1}{4}\operatorname{pr}(F_{m-1}^\bullet)^2,&2m-2\leq n,\\ \operatorname{pr}(F_{m-1}^\bullet),&2m-2>n. \end{cases}$$

Factorization conjecture. There is a decomposition

$$f=\prod_{k=1}^{r'_m+1}g_k^{s_k}$$

where $s_k\in\mathbb{N}^*$ and the $g_k$ are pairwise non-associated, nonconstant elements. This conjecture concerns the factorization pattern responsible for possible failures of polynomiality of the semi-invariant algebra. It is motivated by the cases examined in the paper, but no general proof or resolution is supplied.