Shlyakhtenko–Tao entropy conjecture for the bead process

Let $\mu$ be a probability measure and let $[\mu]_\tau$ denote its free compression for $\tau\in[0,1]$. Define the sub-probability measure $\mu_\tau$ of total mass $\tau$ by $\mu_\tau(A)=\tau[\mu]_\tau(A)$, and let $\sigma_0(u,v)=\log(u)+\log\sin\pi(v/u)$ be the Lagrangian density associated with the quantile function of $(\mu_\tau)_{\tau\in[0,1]}$. Shlyakhtenko and Tao's conjecture. The Lagrangian density $\sigma_0(u,v)$ is proportional to the entropy of the bead process. This conjecture seeks an interpretation of the variational description of free compression in terms of the entropy of the bead process; the source provides no resolution.