Anastos–Fabian–Müyesser–Szabó conjecture for three disjoint perfect matchings

Let $G$ be a graph on $2n$ vertices that is the union of three disjoint perfect matchings. An $(a_1,a_2,a_3)$-matching is a matching containing exactly $a_i$ edges from the $i$th perfect matching. Suppose that $G$ has a component that is not isomorphic to $K_4$. Anastos–Fabian–Müyesser–Szabó conjecture. For any integers $a_1,a_2,a_3\in\mathbb{N}_0$ satisfying

$$a_1+a_2+a_3=n-1,$$

$G$ contains an $(a_1,a_2,a_3)$-matching. This removes the known obstruction formed by a disjoint union of copies of $K_4$ and asks whether that is the only obstruction; the paper reiterates it as an open conjecture.