Hilbert–Schmidt stability of surface groups

Determine whether the surface groups $\pi_1(\Sigma_g)$ associated with closed surfaces of genus $g\geq 2$ are Hilbert–Schmidt stable.

Background

The paper identifies the Hilbert–Schmidt stability of surface groups as unresolved. For a surface group $\pi_1(\Sigma_g)$ with genus $g\geq 2$, the group von Neumann algebra is isomorphic to a free group factor, namely $L(\pi_1(\Sigma_g))\cong L(F_{2g-1})$.

If Hilbert–Schmidt stability were established, the paper’s entropy estimates would imply superexponential representation growth for these groups. The authors note that flexible stability in permutations provides evidence relevant to the question, but does not resolve Hilbert–Schmidt stability.