Brunn–Minkowski conjecture for the alpha-Riesz capacity

Let $N\geq 2$ and let $\alpha\in(0,N)$. For a convex body $K\subset\mathbb{R}^N$, write ${\rm Cap}_{\alpha}(K)$ for its $\alpha$-Riesz capacity. For convex bodies $K_0,K_1$ and $\lambda\in[0,1]$, set $\lambda K_1+(1-\lambda)K_0$ for their Minkowski combination. Brunn–Minkowski conjecture for the $\alpha$-Riesz capacity. The capacity satisfies

$${\rm Cap}_{\alpha}(\lambda K_1+(1-\lambda)K_0)^{\frac{1}{N-\alpha}}\geq \lambda{\rm Cap}_{\alpha}(K_1)^{\frac{1}{N-\alpha}}+(1-\lambda){\rm Cap}_{\alpha}(K_0)^{\frac{1}{N-\alpha}}.$$

This conjecture extends the Brunn–Minkowski inequality proved in the paper for the $1$-Riesz capacity to all $\alpha\in(0,N)$ and would give a fundamental concavity property for Riesz capacity under Minkowski addition.