The free Log-Sobolev inequality at coefficient one-half for the zero potential

Let $Q\equiv0$ on the circle, let $\mu_Q$ be the Haar equilibrium measure, and define

$$I_Q(\mu):=\int(H\mu-Q')^2\,d\mu-\left(\int Q'\,d\mu\right)^2,$$

with

$$H\mu(x)=-\operatorname{p.v.}\int\frac{z+w}{z-w}\,\mu(dw).$$

Write $LSI(\rho)$ for

$$E_Q(\mu)-E_Q(\mu_Q)\leq\frac{1}{4\rho}I_Q(\mu).$$

Zero-potential Log-Sobolev conjecture. For $Q\equiv0$, $LSI(1/2)$ is true. The paper proves that $LSI((1+\delta)/4)$ holds for some $\delta>0$, and conjectures the stronger coefficient $1/2$.