The co-diameter-two co-chordal characterization conjecture

Let $G$ be a co-chordal graph, meaning that its complement is chordal, and suppose that the co-diameter of $G$, namely the diameter of its complement, is $2$. A graph is non-trivially minimally tough if it is minimally tough and not complete. For $\ell\geq 1$, let $S_{\ell,\ell,\ell}$ be obtained from three disjoint stars $K_{1,\ell}$ by adding a triangle between their centers.

Co-diameter-two characterization conjecture. The graph $G$ is non-trivially minimally tough if and only if $G$ is isomorphic to $S_{\ell,\ell,\ell}$ for some $\ell\geq 1$.

This is the formal version of the preceding conjectural description of the unresolved co-diameter-$2$ case for co-chordal graphs, and remains open.