Biane–Speicher's convergence conjecture for a weak double-well potential

Let $V$ be the double-well potential

$$V(x)={1\over 2}x^2+{g\over 4}x^4,$$

where $g$ is a negative constant sufficiently close to zero, and let $\mu_t$ and $\mu_V$ denote the empirical measure process and the equilibrium measure associated with $V$. Biane–Speicher's conjecture. Then $\mu_t$ converges weakly to $\mu_V$. This conjecture concerns convergence for a weak non-convex perturbation of the quadratic potential and is attributed in the source to Biane and Speicher; its general resolution is not stated.