Banuelos and Burdzy's width conjecture for multiple Neumann eigenvalues

Let $\Omega\subseteq\mathbb{R}^2$ be a convex domain. Let $\mu_1(\Omega)$ denote its first nonzero Neumann eigenvalue, let $\mathcal{D}(\Omega)$ be its diameter, and let $\mathcal{W}(\Omega)$ be its width, defined by

$$\mathcal{W}(\Omega)=\min_{v\in\mathbb{S}^{1}}\mathcal{D}(\mathscr{P}_{v}(\Omega)),$$

where

$$\mathscr{P}_{v}(\Omega)=\{x\in\mathbb{R}^2:x\cdot v=0,\ x=y+s\cdot v\ \text{for some}\ y\in\Omega,\ s\in\mathbb{R}\}.$$

Ba~{n}uelos and Burdzy's conjecture. If $\mu_1(\Omega)$ has multiplicity $2$, then

$$\frac{\mathcal{W}(\Omega)}{\mathcal{D}(\Omega)}\geq\frac{1}{\sqrt{2}}.$$

Nadirashvili proved that the multiplicity of the first nonzero Neumann eigenvalue is at most $2$, and Ba~{n}uelos and Burdzy proved simplicity when the width-to-diameter ratio is at most $\frac{\pi}{4j_{0,1}}$. Their conjecture proposes a stronger necessary lower bound for the ratio whenever the first eigenvalue is not simple.