The capacity-deficit asymptotic conjecture for planar shrinking condensers

Let $\Delta$ be the unit disc in the complex plane, let $\mathbb T=\partial\Delta$, and let $E\subseteq\mathbb T$ be closed. For $0<r<1$, write $\operatorname{Cap}(E,\overline{\Delta}(0,r))$ for the condenser capacity, and write $\operatorname{Cap}(E)=\operatorname{Cap}(E,\overline{\Delta}(0,1/2))$. Let $\delta$ denote the deficit of $E$ from full capacity, so that $\operatorname{Cap}(E)=\operatorname{Cap}(\mathbb T)-\delta$. Capacity-deficit asymptotic conjecture.

$$\inf\left\{\operatorname{Cap}(E,\overline{\Delta}(0,r)):\ \operatorname{Cap}(E)=\operatorname{Cap}(\mathbb T)-\delta\right\}\approx\frac{\operatorname{Cap}(\mathbb T)-\delta}{r+\delta}.$$

The conjecture proposes that the relevant asymptotic scale is governed by the small amount by which $E$ fails to have full capacity, rather than by the capacity of a small set. It extends the discussion of the rate at which condenser capacity grows as the inner disc approaches the boundary; the supplied text gives no resolution.