Seymour's improvement conjecture for Hadwiger numbers

Let $G$ be a finite simple graph, let $\alpha(G)$ denote its independence number, and let $\operatorname{had}(G)$ denote its Hadwiger number. Seymour's improvement conjecture. There is a constant $\varepsilon > 0$ such that every graph $G$ with $\alpha(G) \leqslant 2$ satisfies

$$\operatorname{had}(G) \geqslant \left(\frac{1}{3} + \varepsilon\right)|V(G)|.$$

This would improve the classical $|V(G)|/3$ lower bound for graphs with independence number $2$ and is presented as an open conjecture related to Hadwiger's conjecture.