Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets

Let $G$ be a multigraph whose edge-set is the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We also give a construction that shows such a result would be tight. This answers a question of Arman, R\"odl, and Sales about fairly split matchings in the negative.