Aharoni-Berger non-bipartite strong conjecture

Let $M_1,\dots,M_n$ be matchings of size $n+2$ 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 non-bipartite strong conjecture. Every collection of $n$ matchings of size $n+2$ in a graph admits a rainbow matching of size $n$.

The statement is the non-bipartite analogue of the preceding strong conjecture, with two additional edges required in each matching. It was implicitly made in the cited earlier work and stated explicitly in the source, with no resolution supplied here.