The bipartite 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 $a_i$, for $i\in\{1,\ldots,n\}$, be non-negative integers satisfying

$$\sum_{i=1}^n a_i=n-1.$$

Multiplicity Ryser-Brualdi-Stein conjecture. There exists a matching $M$ in $G$ such that

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

This is presented as a multiplicity version of the Ryser-Brualdi-Stein conjecture. It would strengthen the three-colour bipartite conclusion to any number of colours and is open in the source. Noga Alon independently asked this question.