In their 2009 paper (“On a graph property generalizing planarity and flatness”. In: Combinatorica 29.3 (May 2009), pp. 337–361. issn: 1439-6912. doi: 10.1007/s00493-009-2219-6.), van der Holst and Pendavingh defined a new minor monotone graph invariant $\sigma(G)$ for a graph $G$: the minimal integer $k$ such that every CW-complex whose 1-skeleton is $G$ admits a certain even mapping into $\mathbb R^k$.
They were able to prove $\mu(G)\leq\sigma(G)+2$, where $\mu(G)$ is the Colin de Verdière number of $G$ which is minor monotone as well (Colin de Verdière. “Sur un nouvel invariant des graphes et un critère de planaritè”. In: Journal of Combinatorial Theory, Series B 50.1 (1990), pp. 11–21. issn: 0095-8956. doi: 10.1016/0095-8956(90)90093-F.)
My main interest is in the conjecture of van der Holst and Pendavingh in that paper. They conjectured that actually $\mu(G)\leq\sigma(G)$ might hold. Question: What is known about the status of this conjecture? (I have difficulties tracing it as their new invariant $\sigma(G)$ does not seem to have a commonly agreed name yet).