The three-colour exception conjecture for split matchings

Let $G$ be a graph on $2n$ vertices whose edge set is decomposed into perfect matchings $M_1$, $M_2$, and $M_3$, and let $a_1$, $a_2$, and $a_3$ be non-negative integers satisfying

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

A component of $G$ is assumed not to be isomorphic to $K_4$. The three-colour exception conjecture. There exists a matching $M$ in $G$ such that

$$|M\cap M_i|=a_i\qquad\text{for each }i\in\{1,2,3\}.$$

The preceding construction shows that disjoint unions of copies of $K_4$ can obstruct such a matching when all three multiplicities are odd. A positive answer would resolve the three-colour case of the question of Arman, Rödl, and Sales, and would in particular guarantee a matching whenever at least one of the $a_i$ is even, as well as whenever $n$ is odd. The conjecture is open in the source.