Aharoni-Berger strong conjecture for bipartite graphs

Let $M_1,\dots,M_n$ be matchings of size $n+1$ in a bipartite graph. A rainbow matching is a matching whose edges can be assigned distinct indices $i$ so that each edge belongs to $M_i$.

Aharoni-Berger strong conjecture. Every collection of $n$ matchings of size $n+1$ in a bipartite graph admits a rainbow matching of size $n$.

The extra edge in each matching permits a full rainbow matching and implies the preceding bipartite $n-1$ conjecture by adjoining a common dummy edge. The bound $n+1$ is best possible, while the conjecture remains open in the supplied text.