4 problems

Ashbaugh–Benguria-type conjecture. The disk is a solution of the maximisation problem

Non-disk minimiser conjecture. There exist masses $m\geq0$ such that the minimiser of $\lambda_3(\Omega)$ is not the disk.

Krahn–Szegő conjecture. The minimiser of $\lambda_2(\Omega)$ is the union of two disjoint disks, each having area $\frac12$.

Regular polygon conjecture.