  One of the most important open problems in the field of graph colouring or
even graph theory is the conjecture of Hadwiger. This conjecture was the
inspiration for many mathematical works, one of them being the work of
F\"uredi, Gy\'arf\'as and Simonyi in which they "risked" to conjecture the
precise bound for a graph with independence number $2$ to contain a certain
connected matching. We prove that their conjecture would be a corollary of
Hadwiger's conjecture or equivalently if their risky conjecture would be false,
then Hadwiger's conjecture would be false as well.
