Bialy's conjecture on spectral rigidity among ellipses

Let $\rho_0$ and $\rho_1$ be two distinct rotation numbers in $(0,1/2]$. For ellipses $\mathscr E$ and $\mathscr E'$, let $\beta_{\mathscr E}(\rho)$ denote the Mather beta function at rotation number $\rho$. Bialy's conjecture. If

$$\beta_{\mathscr E}(\rho_0)=\beta_{\mathscr E'}(\rho_0),\qquad \beta_{\mathscr E}(\rho_1)=\beta_{\mathscr E'}(\rho_1),$$

then $\mathscr E$ and $\mathscr E'$ are the same up to isometries. This conjecture asks whether two values of Mather's beta function determine an ellipse up to isometry; the source gives no resolution, so the conjecture remains open.