Generalized Mumford conjecture for a nodal-curve moduli space

Let $X_0$ be an irreducible nodal curve with exactly one node, and let $U_{X_0}(2,\mathcal{L}_0)$ be the moduli space of rank-two semistable sheaves on $X_0$ with determinant $\mathcal{L}_0$, where $\mathcal{L}_0$ is an invertible sheaf of odd degree. Embed $X_0$ in a regular family $\pi:\mathcal{X}\to\Delta$ with smooth fibers $\mathcal{X}_s$ for $s\in\Delta^*$, and let $P_k^{\mathrm{mon}}$ be the monodromy-invariant subspace of the primitive component $P_k$ of $H^*(M_{\mathcal{X}_s}(2,\mathcal{L}|_{\mathcal{X}_s}),\mathbb{Q})$. Generalized Mumford conjecture. The space $P_k^{\mathrm{mon}}$ is independent up to isomorphism of the choice of the family $\pi$, and

$$H^*(U_{X_0}(2,\mathcal{L}_0),\mathbb{Q})\cong\bigoplus\limits_{k=0}^g P_k^{\mathrm{mon}}\otimes\mathbb{Q}[\alpha,\beta,\gamma]/I_{g-k}.$$

This proposes that the Mumford decomposition for smooth curves extends to the singular fixed-determinant moduli space through monodromy-invariant primitive pieces. The supplied text gives no resolution, so the conjecture remains open here.