Large-nilradical component conjecture for moduli of sheaves on ribbons

Let $X$ be a ribbon, let $\delta=-\deg({\mathscr N})$, and let $\bar{g}$ be the genus of $X_{\mathrm{red}}$. The large-nilradical component conjecture. If

$$\delta>2\bar{g}-2,$$

then the only irreducible components of the moduli space of coherent sheaves on $X$ are those whose generic elements are quasi locally free sheaves of rigid type when the generalized rank is odd, or generalized vector bundles when the generalized rank is even. This is motivated by deformation results for vector bundles on the reduced curve and by the generization conjecture, but the asserted description is not established in general.