Volume-normalized p-capacity conjecture for linear images

Let $n\geq 2$ and $0<p<n$. Let $K\subset\mathbb{R}^n$ be a compact set with an irreducible group of isometries, let $M$ be a volume-preserving invertible linear map, and write $C_p$ for $p$-capacity and $V$ for volume. Volume-normalized $p$-capacity conjecture. One should have

$$C_p(MK)\geq C_p(K).$$

In particular, if $K$ has positive volume, then the scale-invariant quantity $C_p/V^{1/n}$ should be minimal at $K$ among all linear images:

$$\frac{C_p}{V^{1/n}}(MK)\geq\frac{C_p}{V^{1/n}}(K).$$

This is posed as an open volume-normalization analogue of the paper's established capacity-minimization results for other settings; its validity for general $p$ is left open.