Finite area determination conjecture for Kähler–Einstein metrics on a product of surfaces

Let $\Sigma$ be the closed surface under consideration, let $\mathcal L^*(g)$ denote the relevant set of surface subgroups of genus $g$, and let $A(h)(\Pi)$ be the associated area function of a metric $h$ on $\Sigma\times\Sigma$. Finite area determination conjecture. There is a finite set $\mathcal L_0\subset\mathcal L^*(g)$ such that, if $h$ and $\hat h$ are Kähler–Einstein metrics on $\Sigma\times\Sigma$ satisfying

$$A(h)(\Pi)=A(\hat h)(\Pi)\quad\text{for all }\Pi\in\mathcal L_0,$$

then $h$ and $\hat h$ are isometric, and the isometry is homotopic to the identity. This asks whether finitely many minimal-Lagrangian area measurements determine a Kähler–Einstein metric up to an isometry homotopic to the identity; the supplied text presents it as a new question, with no resolution stated.