Rigidity conjecture for polarized abelian varieties from Bergman densities

Let $(X,L)$ and $(X',L')$ be polarized abelian varieties of dimension $n$. Let $\rho_{L,k}$ and $\rho_{L',k}$ denote their Bergman densities at level $k$, and let $\rho_{L,k}=\rho_{L,k}$ be understood on $X$ and $X'$ via pullback by a diffeomorphism. Suppose there is a diffeomorphism $\Phi:X\to X'$ and an integer $k\geq 1$ such that

$$\rho_{L,k}=\Phi^*\rho_{L',k}.$$

Rigidity conjecture. Then $\Phi$ is either a biholomorphism or an anti-biholomorphism, and moreover

$$\Phi^*((L')^{\!k})\cong L^{\!k}\qquad\text{or}\qquad \Phi^*((L')^{\!k})\cong\overline{L^{\!k}},$$

respectively. Here $\overline{L}$ denotes the complex-conjugate line bundle of $L$, whose transition functions are given by complex conjugates.

This conjecture asserts that a single Bergman density determines the complex structure and the relevant tensor power of the polarization up to holomorphic or anti-holomorphic equivalence. Its status is not resolved in the supplied text.