Free probability via entropic optimal transport

Let $μ$ and $ν$ be compactly supported probability measures on $\mathbb{R}$, and let $μ\boxplus ν$ denote their additive free convolution. We show that for all sufficiently large real $z$, \begin{equation} \int_{-\infty}^\infty \log(z - x) \, (μ\boxplus ν)(\mathrm{d}x) = \sup_Π \left\{ \mathbf{E}_Π[