The strip energy ball comparison conjecture

Let $1 \leq q < n$. Suppose $K \subset \mathbb{R}^n$ is compact and $B \subset \mathbb{R}^n$ is a closed ball. The quantities $E_K(t)$ and $E_B(t)$ denote the strip energies of $K$ and $B$ at height $t>0$.

Strip energy ball comparison conjecture. If

$$\lim_{t \to \infty} E_K(t) = \lim_{t \to \infty} E_B(t),$$

then

$$E_K(t) \leq E_B(t)$$

for all $t>0$.

This is proposed as a stronger conjecture related to the open Pólya–Szegő conjecture comparing logarithmic and Newtonian capacities. Establishing the comparison would provide a framework for resolving that classical capacity inequality.