4 problems

Let $u{\phi}$ and $u{\psi}$ be free Gibbs measures associated with strictly $\mathcal{C}^2$ convex potentials $\phi,\psi:\mathbb{R}rightarrowmathbb{R}$, and write $mu=nu{\phi}$ and…

Eventual unimodality conjecture. The free convolution powers $\mu^{\boxplus s} are unimodal for every$sgeq T$.

Discontinuity conjecture. The preceding assertion holds for every such probability measure $u$.

Hinz–Młotkowski conjecture. The double power $\tilde{\bm{\pi}}_{st}$ is a probability measure if and only if $\max(s,t)\geq1$. Equivalently, the sequence…