Prove the interval conjecture for capacity ratios

Prove that for \(0<r<s<1\), the closed interval \([-1,1]\) minimizes \(\operatorname{Cap}_{-s}(K)/\operatorname{Cap}_{-r}(K)\) among all compact subsets \(K\subset\mathbb R\) containing at least two points.

Background

For negative exponents, the paper writes $r=-p$ and $s=-q$, so the capacity ratio becomes $$\operatorname{Cap}_{-s}(K)/\operatorname{Cap}_{-r}(K)$$. The interval conjecture concerns the upper triangle $0$-1<p<q<0$. Numerical experiments with finite subsets of the line support the conjecture, but the paper does not establish the required inequality for all compact sets.

The authors note that the interval capacity is explicitly computable and that scaling invariance makes all nondegenerate bounded intervals equivalent for the ratio. The unresolved issue is the global comparison with arbitrary compact subsets of the real line.