Maximum matching M with the condition G[M] is 2K_2-free Is there anything in the literature close to the following problem:

Given a bipartite graph $G(V,E)$ with balanced bipartition $ \{U,W\}$ , does there exist a perfect matching $ M $ in $ G $ such that for every 2 edges $u_1w_1, u_2w_2\in M $, there is an edge $u_1w_2$ or edge $u_2w_1$ (or both) in $ G $?

In other words, is there a perfect matching $M$ such that the induced subgraph $G[M]$ is $ 2K_2 $-free.
(With balanced bipartition, I meant $|U|=|W|$.)

The extra condition is something like an opposite extreme of that used in induced matching problem. Another possibly related one is the problem of finding maximum size matching $M$ in bipartite graph $G$ such that contraction of edges in $M$ minimizes the number of edges left in the graph.

I checked the list of matching related problems given by Plummer in Matching and vertex packing: how "hard" are they? without success.

PS: This problem is a special case of this decision problem :- For a given $k\in\mathbb{N}$, is there a maximum matching $M$ of a bipartite graph $G$ such that $G[M]$ is $2K_2$-free and $|M|&gt;k$. If the input graph is balanced bipartite and $k=|U|$, we get the above problem.

Thank you.
Surprise! (for me).
This type of matchings are already studied in the literature. They are called connected matchings.

They were introduced by Plummer, Stiebitz and Toft in their study on Hadwiger conjecture. See the chapter "Connected Matchings" by Cameron in the book "Combinatorial Optimization – Eureka, You Shrink!"

The status of connected matchings in bipartite graphs (not necessary balanced) is open to the best of my knowledge (i shall update). The weighted version of the problem is NP-complete for bipartite graphs. The problem is polynomial time solvable for chordal bipartite graphs.

Update: the problem is NP-complete for balanced bipartite graphs (i.e., the exact problem asked in the question). This is proved in the paper "Multitasking Capacity: Hardness Results and Improved Constructions" by Alon et al.  They also report that finding the size of a largest connected matching is hard to approximate within a factor of $n^{1-\epsilon}$ unless NP = co-RP.

Notes added earlier (kept for interested people):
"Connected matchings in chordal bipartite graphs" by Jobson et al. (doi:https://doi.org/10.1016/j.disopt.2014.06.003) and "Connected matchings in special families of graphs" by Caragianis (thesis) are two notable references.