Convex-function moment inequality for conformal-centroid compact continua

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $\mu_K$ be the equilibrium measure of $K$, and let $L=[-2,2]$ with equilibrium measure $\mu_L$. Convex-function moment conjecture. For every $\phi\in C^1(\mathbb{R})$ such that both $\phi$ and $\phi'$ are convex,

$$\int_K\phi(\log|z|)\,d\mu_K\leq\int_L\phi(\log|z|)\,d\mu_L.$$

The source presents this as an equivalent formulation of the averaged conjecture; it is not stated as resolved.