Reverse Faber–Krahn conjecture for the logarithmic potential operator

Let $\Omega$ be a bounded domain, let $\Omega^*$ denote the comparison domain of the same area, and let $B_R$ be the comparison ball for a domain whose transfinite diameter is $R$. The reverse Faber–Krahn conjecture.

$$\tilde{\tau}_1(\Omega)\leq \tilde{\tau}_1(\Omega^*).$$

Additionally, for a domain $\Omega$ with transfinite diameter $R$,

$$\tilde{\tau}_1(\Omega)\leq \tilde{\tau}_1(B_R).$$

These are proposed as reverse Faber–Krahn inequalities for the logarithmic potential operator; their status is open.