The maximal-gamma conjecture for minimal graphs

Let $\ell$ be a positive integer, let $G$ be an $\ell$-minimal graph, let $a$ be the vector used in the definition of $\gamma_{\ell-1}(G,a)$, and let $\bar{e}$ denote the all-ones edge vector. Let $\hat{\mathcal{K}}_{\ell+2,\ell-1}$ denote the specified class of sparse stretched cliques.

Maximal-gamma conjecture. For every positive integer $\ell$, the maximum value of $\gamma_{\ell-1}(G,a)$ among $\ell$-minimal graphs is attained by a sparse stretched clique $G \in \hat{\mathcal{K}}_{\ell+2,\ell-1}$ and $a \coloneqq \bar{e}$.

The observed 4-minimal examples motivate this conjecture, which predicts both the extremal graph class and the maximizing vector. No general proof or disproof is supplied in the source.