Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite.
Now, is there a way to choose disjoint matchings from the three bipartitie subgraphs such that the union of the three disjoint matchings yields us two disjoint maximal matchings of $G$. We could easily get one maximal matching by the union of three disjoint matchings obtained from the three distinct bipartite subgraphs, but getting two maximal matchings seems hard to me at present. Maybe, we must use some symmeteric difference of two matchings. But, anyways, it is unclear. Any hints? What if the graph $G$ were 1-factorizable, or, Class $1$? Thanks beforehand.