The multiplicity-two conjecture for the second Dirichlet eigenvalue on simply connected planar domains

Let $\Omega\subset\mathbb{R}^2$ be a bounded simply connected domain, and let $\lambda_2(\Omega,-\Delta,\mathfrak{d})$ be its second Dirichlet eigenvalue. Write $\operatorname{mult}(\lambda_2;\Omega,-\Delta,\mathfrak{d})$ for the multiplicity of this eigenvalue.

Multiplicity-two conjecture.

$$\operatorname{mult}(\lambda_2;\Omega,-\Delta,\mathfrak{d})\leq 2$$

for any simply connected bounded domain $\Omega$.

The bound is known for convex domains and domains for which the nodal line conjecture holds. It is motivated by the sharp multiplicity bound $3$ for general non-simply-connected planar domains, and remains open in the stated generality.