Ramachandran's Bochner–Hartogs or surface-group dichotomy

Let $X$ be a smooth projective variety with a type B Zariski-dense representation $\rho:\pi_{1}(X)\longrightarrow G$ to a Lie group $G$ defined over a local field $K$, and set $H=\ker(\rho)$. Assume that $H$ is finite. The Bochner–Hartogs property for $\widetilde{X}$ means that for every compactly supported $(0,1)$-form $\alpha\in A^{0,1}_{c}(\widetilde{X})$ with $\bar\partial\alpha=0$, there is a compactly supported smooth function $u\in C^{\infty}_{c}(\widetilde{X})$ such that $\bar\partial u=\alpha$.

Ramachandran's conjecture. One of the following holds: the universal cover $\widetilde{X}$ has the Bochner–Hartogs property, or $\pi_{1}(X)$ is commensurable with the fundamental group of a compact Riemann surface.

The paper attributes this question to M. Ramachandran and presents it as a possible application of the Shafarevich–Kollár conjecture. The supplied text gives no resolution status.