Wide Young diagram matching conjecture

For a hypergraph $H$, a 2-matching is a set of edges such that every two edges share fewer than two vertices, and let $\nu^{(2)}(H)$ denote its maximum size. For a Young diagram $Y$, let $H(Y)$ be the tripartite hypergraph whose sides are the rows, columns, and numerical symbols, with edges $r_i c_j s_k$ whenever $1\leq j,k\leq a_i$. Write $|Y|$ for the number of squares of $Y$. Wide Young diagram matching conjecture. If a Young diagram $Y$ is wide, then

$$\nu^{(2)}(H(Y))=|Y|.$$

The source presents this as an equivalent formulation of the WPC. Since a 2-cover of $H(Y)$ of size $|Y|$ is immediate, the conjecture asserts the matching-side equality corresponding to the existence of a Latin filling; its resolution status is not specified in the supplied text.