Botler et al.'s bipartite minor conjecture for independence number two

Let $G$ be an $n$-vertex graph with independence number $\alpha(G)$. For positive integers $\ell$ and $r$, let $K_{\ell,r}$ denote the complete bipartite graph with parts of sizes $\ell$ and $r$.

Botler et al.'s conjecture. Let $G$ be an $n$-vertex graph with $\alpha(G)\leq2$. For any positive integer $\ell$ with $\ell<\lceil n/2\rceil$, we have

$$G\succeq_m K_{\ell,\lceil n/2\rceil-\ell}.$$

The source describes this as slightly weaker than the Plummer–Stiebitz–Toft reformulation. It remains open in the generality stated, while the paper proves a related stronger-looking result for the graphs under consideration.