The Bergman density rigidity conjecture for polarized hyperbolic Riemann surfaces

Let $(X_i,\omega_i,L_i,h_i)$, $i=1,2$, be polarized hyperbolic Riemann surfaces as in the setting of the paper's main theorem, and let $\rho_{X_i,k}$ denote their Bergman density functions. Suppose that $\Phi:X_1\to X_2$ is a diffeomorphism satisfying, for some $k\geq 5$,

$$\Phi^*\rho_{X_2,k}=\rho_{X_1,k}.$$

Bergman density rigidity conjecture. Then $\Phi$ is either holomorphic and

$$\Phi^*L_2^k=L_1^k,$$

or anti-holomorphic and

$$\Phi^*L_2^k=\overline{L_1^k}.$$

This conjecture proposes that a single Bergman density at sufficiently high level determines the polarized hyperbolic Riemann surface up to holomorphic or anti-holomorphic equivalence. The source gives no resolution status.