The chromatic bound conjecture for the first Steklov eigenvalue

Let $\Sigma$ be a compact surface with boundary. Write $m_1(\Sigma)$ for the maximal multiplicity of the first Steklov eigenvalue $\sigma_1(\Sigma,g,\rho,\gamma)$ as the metric $g$ and the positive densities $\rho$ and $\gamma$ vary. Let $\mathrm{Chr}_0(\Sigma)$ be the supremum of the chromatic numbers of finite graphs admitting a proper embedding in $\Sigma$, where proper means that every vertex is mapped to $\partial\Sigma$. Chromatic bound conjecture. For every compact surface $\Sigma$ with boundary,

$$m_1(\Sigma)=\mathrm{Chr}_0(\Sigma)-1.$$

This conjecture relates the multiplicity of the first Steklov eigenvalue to the relative chromatic number of the surface. The supplied text gives no evidence of a resolution.