Clustered chromatic conjecture for Colin de Verdière classes

Let $\mathcal{V}_k:=\{G:\mu(G)\leqslant k\}$, where $\mu(G)$ is the Colin de Verdière parameter, and let $\chi_{\star}(\mathcal{V}_k)$ be the clustered chromatic number of this class. Clustered Colin de Verdière conjecture. For every integer $k\geqslant 1$,

$$\chi_{\star}(\mathcal{V}_k)=k+1.$$

The defective analogue is known, and clustered colourings are presented as a natural approach to Colin de Verdière's chromatic conjecture; the asserted equality is not resolved in the supplied text.