Free Cramér conjecture for strictly convex free Gibbs measures

Let $u_{\phi}$ and $u_{\psi}$ be free Gibbs measures associated with strictly $\mathcal{C}^2$ convex potentials $\phi,\psi:\mathbb{R}\rightarrow\mathbb{R}$, and write $\mu=\nu_{\phi}$ and $\nu=\nu_{\psi}$. The free convolution $\mu\boxplus\nu$ is defined by free independence; a probability measure is semicircular when it is a semicircular law. Weak free Cramér conjecture. If $\mu\boxplus\nu$ is semicircular, then both $\mu$ and $\nu$ are semicircular. This is motivated by the failure of the classical Cramér theorem in the free setting and would provide a weak free analogue of Cramér's theorem; it remains open in the source.