Multiplicity Ryser-Brualdi-Stein conjecture

Let $G$ be a complete bipartite graph on $2n$ vertices whose edge set is decomposed into perfect matchings $M_i$, for $i=1,\ldots,n$. Let $a_1,\ldots,a_n\in\mathbb{N}_0$ be nonnegative integers satisfying

$$\sum_i a_i=n-1.$$

A matching with prescribed multiplicities is a matching whose intersection with each $M_i$ has the specified size. Multiplicity Ryser-Brualdi-Stein conjecture. There exists a matching $M$ in $G$ such that

$$|M\cap M_i|=a_i$$

for every $i\in\{1,\ldots,n\}$. The conjecture extends the rainbow matching formulation by prescribing how many edges are selected from each color class; the paper presents it as an open conjecture and proves the three-color case.