Let $\Gamma = (M, W, K)$ be a bipartite graph, that is $M, W$ are sets and $K\subseteq M\times W$. If there is an injective function $f:M\to W$ such that $f\subseteq K$ we say $f$ is an espousal and $\Gamma$ is espousable. Moreover, $\Gamma$ is critical, if it is espousable, and every espousal is surjective.
For a bipartite graph $\Gamma = (M, W, K)$ and $A\subseteq M$ we set $W[A] = \{w \in W: (\exists m\in A): (m,w)\in K\}$. We also set $\Gamma|_A = (A, W[A], K\cap(A\times W[A])$
If $\kappa$ is a cardinal then we say that $\Gamma = (M, W, K)$ is a $\kappa$-impediment if there is a subset $A\subseteq M$ with $|A| = \kappa$ and $\Gamma|_{M\setminus A}$ is critical.
Question: Let $\Gamma = (M, W, K)$ be a 1-impediment and let $M$ be partitioned into $M_1\cup M_2$ and $W$ be partitioned into $W_1\cup W_2$. Is it true that at least one of $(M_i, W_i, K\cap (M_i\times W_i))$ contains a $\kappa$-impediment for $\kappa \leq \aleph_0$?
