Odd minor analogue for graphs with independence number two

Let $G$ be a finite simple graph with independence number $\alpha(G)$ and chromatic number $\chi(G)$. For positive integers $\ell$ with $2\ell\leq\chi(G)$, let $K^{\ell}_{\ell,\chi(G)-\ell}$ denote the graph used in the source: it is obtained from the disjoint union of a complete graph $K_\ell$ and an independent set on $\chi(G)-\ell$ vertices by adding all possible edges between them. Write $G\succeq_{om}H$ when $H$ is an odd minor of $G$. Odd minor analogue for graphs with independence number two. If $G$ is a graph with $\alpha(G)\leq2$, then, for every positive integer $\ell$ with $2\ell\leq\chi(G)$, we have

$$G\succeq_{om}K^{\ell}_{\ell,\chi(G)-\ell}.$$

The source presents this as a conjectured odd-minor analogue of a corresponding minor result; no resolution is given.