In the normal version of the matching problem, we are given a set of vertices $X$, $Y$, and $Z$, each of size $n$, and a set of edges $E\subseteq X\times Y\times Z$. We need to find a matching $M\subseteq E$ s.t. $|M| = n$ and it covers each element of $X$, $Y$, and $Z$ exactly once.
The variation is, can we find $M$ s.t. for any $(x_i,y_i,z_i), (x_j,y_j,z_j) \in E\smallsetminus M$, $x_i = x_j$ or $y_i = y_j$ or $z_i = z_j$?
This is the decision version of the 3d-matching variant. I wanted to know if this would also be NP-complete.