The clique-rank bound conjecture for stretched-clique classes

Let $\ell$ be a positive integer. Let $\mathcal{K}_{\ell+2,\ell-1}$ be the specified graph class, let $\omega(G)$ denote the clique number of $G$, and let $r_+(G)$ denote its Lovász–Schrijver positive-semidefinite rank.

Clique-rank bound conjecture. For every positive integer $\ell$, if $G \in \mathcal{K}_{\ell+2,\ell-1}$ and $\omega(G) \geq 4$, then

$$r_+(G) \leq \ell-1.$$

The paper verifies the analogous phenomenon for the examined 4-minimal setting by showing that relevant graphs with clique number at least four have rank at most three. The extension to every positive integer $\ell$ is conjectural.