Analytic embedding conjecture for Bergman spaces on non-compact Riemann surfaces

Let $M$ be a non-compact Riemann surface together with an analytic embedding

$$M \hookrightarrow {\mathbb C}^n.$$

Analytic Bergman-space conjecture. The statement of the smooth affine algebraic-curve theorem holds for $M$: for every open subset $U\subseteq M$, the Bergman space $A^2(U,\sigma)$ is either infinite-dimensional or trivial. This proposes an analytic analogue of the algebraic result, whose proof relies essentially on algebraicity; the source gives no resolution.