Boundary Colin de Verdière conjecture for the first Steklov eigenvalue

Let $M$ be a compact surface with boundary, and let $\sigma_1(M)$ denote its first nonzero Steklov eigenvalue. Define $\mathrm{Chr}(M,\partial M)$ to be the number of vertices of the largest complete graph that can be embedded in $M$ with all vertices on $\partial M$. Boundary Colin de Verdière conjecture. The maximal multiplicity of $\sigma_1(M)$ is

$$\mathrm{Chr}(M,\partial M)-1.$$

This conjecture is a Steklov analogue of Colin de Verdière's conjecture for Schrödinger operators on surfaces, replacing the chromatic number by a boundary-embedded graph invariant. The supplied source does not report a resolution.