Fixed-locus Euler characteristic conjecture for invariant compactified Jacobians

Let $A$ be an Abelian surface of Picard rank one, let $\tau$ be the involution defining the quotient, and let $C\subset A$ be an integral $\tau$-invariant curve of geometric genus $g$ whose quotient by $\tau$ is rational. Write $\operatorname{\overline{Jac}}(C)$ for the compactified Jacobian of $C$, and let $\operatorname{\overline{Jac}}(C)^{\tau}$ denote its $\tau$-fixed locus. Fixed-locus Euler characteristic conjecture. Then

$$e\bigl(\operatorname{\overline{Jac}}(C)^{\tau}\bigr)=2^{2g}.$$

The conjecture would explain the contribution of each such invariant curve to the enumerative geometry of the orbifold Kummer surface. The source presents the identity as being supported by the Picard-rank-one comparison of the relevant generating functions, but gives no resolution.