The chromatic lower-bound conjecture for the Colin de Verdière parameter

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $\nu(G)$ denote the positive semidefinite Colin de Verdière parameter.

Chromatic lower-bound conjecture. For any graph $G$,

$$\nu(G)\geq \chi(G)-1.$$

This would follow from either the $\delta$-conjecture or Hadwiger's conjecture, since the source explains that each would imply the required lower bound. The inequality is stated as open in the source; current results supplied there give only a weaker bound proportional to $\chi(G)$.