Averaged 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 $g$ be the Green's function of $K^c$ with pole at infinity, let $L=[-2,2]$, and let $J(r,K)=\int_r^R I(t,K)\,\frac{dt}{t}$, where $I(r,K)=\frac{1}{2\pi}\int_{-\pi}^{\pi}g(re^{i\theta})\,d\theta$. Averaged moment conjecture. For every $R\geq2$,

$$J(r,K)\leq J(r,L),\qquad r\in[0,R].$$

This is proposed as an averaged substitute for the moment inequality that holds under symmetry but fails under the weaker conformal-centroid condition; its status is not resolved in the source.