Reverse Faber–Krahn conjecture for the negative logarithmic eigenvalue

Let $\Omega$ be a bounded domain and let $H$ be a polarizer. The reverse Faber–Krahn conjecture.

$$\widetilde{\tau}_1(\Omega)\leq \widetilde{\tau}_1(P_H(\Omega)).$$

The inequality is immediate in the cases discussed in the source when the transfinite diameters place the domains on opposite sides of the positivity threshold, but it remains open in general.