The completely reducible matrix-polynomial conjecture for canonical compactified Jacobians

Let $C$ be a nodal curve satisfying the condition referred to as. Let $(\pazocal P_C^B)_{\mathrm{cr}}$ denote the locus of completely reducible matrix polynomials, let $G^B$ be the centralizer of $B$, and let $\pazocal P_{m,n}$ be the space of $n\times n$ matrix polynomials of degree $m$. For $P\in\pazocal P_{m,n}$, write $C_P$ for its spectral curve. The moduli space

$$(\pazocal P_C^B)_{\mathrm{cr}} / G^B = \{P \in \pazocal P_{m,n} \mid P \text{ is completely reducible},\ C_P = C \} \,/\, \operatorname{GL}_n(\mathbb{C})$$

Completely reducible matrix-polynomial conjecture. This moduli space, consisting of completely reducible $n\times n$ degree $m$ matrix polynomials with spectral curve $C$, considered up to conjugation by constant matrices, is a variety isomorphic to an open dense subset of the canonical compactified Jacobian of $C$.

The conjecture proposes that the quotient by the conjugation action of the centralizer identifies the completely reducible matrix-polynomial moduli with the canonical compactified Jacobian up to an open dense subset. It is motivated by the agreement between the combinatorial stratifications of the completely reducible compactification and Alexeev's canonical compactified Jacobian, while their individual strata are initially described using different Jacobian-type spaces.