Two-dimensional inequality without topological restrictions

Determine whether the inequality \(\mu_3(\Omega)<\lambda_1(\Omega)\) holds for every bounded two-dimensional domain \(\Omega\) satisfying the stated compactness condition, without imposing topological restrictions on the domain.

Background

The paper establishes $\mu_3<\lambda_1$ in dimensions $d\geq 3$, and proves the same inequality in two dimensions for bounded doubly connected Lipschitz domains that are symmetric with respect to the origin. It also recalls that the inequality was previously known for simply connected planar domains. The remaining issue is whether the result extends to arbitrary two-dimensional domains, including multiply connected domains without the symmetry and boundary assumptions used in Theorem 1.2.