Take $N \in \mathbb{N}$.
If a have $2N$ points distributed uniformly at random inside the unit square, and construct the minimum weight triangulation $\mathcal{T}_{2N}$ on those points (edge weights are the Euclidean lengths of the edges), does it contain the minimum weight perfect matching $\mathcal{M}_{2N}$?
"Flipping" triangles whose union forms the minimum triangulation on the points, and whose edges contain the minimum matching on that triangulation, allows for new matchings which are only ever longer than the one already found, so I imagine the answer is affirmative?