The independence-number lower-bound conjecture for the Colin de Verdière parameter

Let $G$ be a graph with $n$ vertices, let $\alpha(G)$ denote its independence number, and let $\nu(G)$ denote the positive semidefinite Colin de Verdière parameter.

Independence-number lower-bound conjecture. For any graph $G$,

$$\nu(G)\geq \frac{n}{\alpha(G)}-1.$$

This is a weaker consequence of the chromatic lower-bound conjecture, using $\chi(G)\geq n/\alpha(G)$. The source says that it is not known in general.