Holomorphic equivalence conjecture for isometric Bergman spaces

Let $D_1$ and $D_2$ be $A^p$-complete bounded domains for some $p\in(0,2)$. A linear isometry between their Bergman spaces $A^p(D_1)$ and $A^p(D_2)$ is a linear map preserving the $A^p$-norm.

Holomorphic equivalence conjecture. If there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$, then $D_1$ and $D_2$ are holomorphically equivalent.

For $p\in(0,2)$, the conjecture proposes that the condition of not being of boundary blow down type in the paper's main theorem is unnecessary. It concerns when the geometry of bounded domains is determined by the linear-isometric structure of their Bergman spaces.