Zaks's conjecture on perfect matchings in connected graphs

Let $k \ge 3$ and $n \ge 2k$ be integers, and let $G$ be a matchable $k$-connected graph with $n$ vertices. Zaks's conjecture. Every such graph satisfies

$$\Phi(G) \ge k!;$$

moreover, infinitely many graphs attain equality. The conjecture is disproved by the counterexamples constructed in the paper.