Second covering allocation conjecture for Young diagrams

Let $Y$ be a Young diagram and let $H(Y)$ be the associated 3-partite 3-uniform hypergraph. Write $|Y|$ for the number of cells of $Y$, let $\tau^{(2)}(H(Y))$ denote its second covering number, and let an allocation be the coarse filling notion used in the paper. Second covering allocation conjecture. If

$$\tau^{(2)}(H(Y))=|Y|,$$

then $Y$ has an allocation. The paper states that this is equivalent to the Wide Partition Conjecture and the allocation conjecture. A weaker covering-number equality is known for wide diagrams, but the implication to an allocation remains open.