Stability-independence conjecture for the cohomology rings of compactified Jacobians

Let $C_0$ be the central curve in the setting of Theorem 0.8, and let $\phi$ and $\phi'$ be nondegenerate stability conditions. Write $\overline{J}^{\phi}_{C_0}$ and $\overline{J}^{\phi'}_{C_0}$ for the corresponding compactified Jacobians. Stability-independence conjecture. Under the assumption of Theorem 0.8, there is an isomorphism

$$H^*(\overline{J}^{\phi}_{C_0},\mathbb{Q})\simeq H^*(\overline{J}^{\phi'}_{C_0},\mathbb{Q})$$

of graded $\mathbb{Q}$-algebras. This conjecture predicts that, although the ordinary cohomology ring can depend sensitively on the stability condition in general, the rings for these central compactified Jacobians become independent of the choice of nondegenerate stability condition under the stated hypothesis.