The strong $\delta$-conjecture for the Colin de Verdière parameter

Let $G$ be a graph. Let $\delta(G)$ be its minimum degree, let $l(G)$ be its degeneracy,

$$l(G)=\max\\{\delta(H):H\text{ is a subgraph of }G\\},$$

let $\lceil\delta\rceil(G)$ be the maximum of $\delta(H)$ over minors $H$ of $G$, and let $\nu(G)$ be the positive semidefinite Colin de Verdière parameter.

Strong $\delta$-conjecture for $\nu$. For any graph $G$,

$$\delta(G)\leq l(G)\leq \lceil\delta\rceil(G)\leq \nu(G).$$

The first two inequalities follow directly from the definitions, while the conjectural content is the final inequality. The source later says that the corresponding original $\delta$-conjecture is used conditionally and does not supply a resolution of this full chain in the candidate span.