The sparsest minimal-graph conjecture

For a positive integer $\ell$, let an $\ell$-minimal graph mean a graph that is minimal at Lovász–Schrijver rank $\ell$. Let $\hat{\mathcal{K}}_{\ell+2,\ell-1}$ denote the specified class of sparse stretched cliques.

Sparsest minimal-graph conjecture. For every positive integer $\ell$, a sparsest $\ell$-minimal graph is a sparse stretched clique in $\hat{\mathcal{K}}_{\ell+2,\ell-1}$.

The conjecture generalizes the observed structure of the sparsest 4-minimal graphs, all 40 of which were found to be sparse stretched cliques in $\hat{\mathcal{K}}_{6,3}$. Its validity for arbitrary positive integer $\ell$ remains open.