Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ is Gorenstein ($K_X$ is Cartier) and $R^1\pi_* \mathcal{O}_Y=\mathbb{C}_x$ where $\pi:Y\to X$ is a resolution of the singularity.

Equivalently: $\pi_*\omega_Y=m_x\omega_X$.

Thanks in advance, Benoit