Aharoni-Berger conjecture for graphs

Let $M_1,\dots,M_n$ be matchings of size $n$ in a 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 conjecture. Every collection of $n$ matchings of size $n$ in a graph admits a rainbow matching of size $n-1$.

This drops the bipartiteness requirement from the preceding conjecture. The $n-1$ bound is best possible, as shown by the cycle construction described in the source, but the general assertion remains open in the supplied text.