Wiegerinck-type conjecture for $L^2$ holomorphic sections on compact Riemann surfaces

Let $M$ be a compact, connected Riemann surface, let $E\rightarrow M$ be a holomorphic vector bundle with a smooth Hermitian metric $h$, and let $dV$ be a smooth volume form. For a domain $D\subset M$, define

$$\mathcal O L^2(\left.E\right|_D)=\left\{s\in H^0(D,E):\int_D h(s,s)\,dV<\infty\right\}.$$

The space $H^0(M,E)$ of global holomorphic sections is finite dimensional and is contained in $\mathcal O L^2(\left.E\right|_D)$. Wiegerinck-type conjecture.

$$\mathcal O L^2(\left.E\right|_D)\text{ is either equal to }H^0(M,E)\text{ or has infinite dimension}.$$

This is proposed as a natural generalization of Wiegerinck's theorem from functions on planar domains to holomorphic sections of vector bundles over domains in compact Riemann surfaces. The supplied text does not state whether it has been proved or remains open.