Support rigidity conjecture for Liouville currents of flat metrics

Let $S$ be the underlying surface and let $\mathrm{Flat}(S)$ denote the space of flat metrics on $S$. For $\varphi\in\mathrm{Flat}(S)$, let $L_{\varphi}$ be its Liouville geodesic current and let $\mathrm{supp}(L_{\varphi})$ denote its support. Scaling a metric by $a>0$ preserves this support, and affine deformations arising from $SL_2(\mathbb R)$ actions on metrics defined by holomorphic quadratic differentials also preserve it. Support rigidity conjecture. Given $\varphi_1,\varphi_2\in\mathrm{Flat}(S)$, if

$$\mathrm{supp}(L_{\varphi_1})=\mathrm{supp}(L_{\varphi_2}),$$

then there exists an affine map $f\colon(S,\varphi_1)\to(S,\varphi_2)$ isotopic to the identity. Equivalently, up to this affine ambiguity, the support of the Liouville current determines the flat metric. The paper proves that the full marked length spectrum determines a flat metric, and studies how much information is already contained in the support of its Liouville current. The conjecture asserts that the scaling and affine deformations described in the open questions are the only possible ambiguities.