Füredi–Gyárfás–Simonyi conjecture on connected matchings

Let $G$ be a graph on $4t-1$ vertices with independence number $\alpha(G)=2$. A connected matching is a matching whose edges are pairwise linked by edges of $G$; write $\operatorname{cm}(G)$ for the size of a largest connected matching.

Füredi–Gyárfás–Simonyi conjecture.

$$\operatorname{cm}(G)\geq t.$$

This conjecture is a precise version of the problem of determining the largest connected matching in graphs with independence number two. The paper reports that it is known for $t\leq22$, while the general case remains open; a counterexample would also imply a counterexample to Hadwiger's conjecture.