The partite-structure conjecture for packed hypergraphs

Let $H$ be a hypergraph with edge ideal $I=I(H)$. Write $\alpha(I)$ for the least degree of a nonzero homogeneous element of $I$, and let $\operatorname{ht}(I)$ denote the height of $I$. A hypergraph is $(a:b)$-partite when it has the specific partite structure used to obtain the Waldschmidt lower bound $\widehat{\alpha}(I)\geq a/b$. The partite-structure conjecture. If $H$ has the packing property, then $H$ is

$$(\alpha(I)\operatorname{ht}(I):\operatorname{ht}(I))\text{-partite}.$$

The proposed structure would apply the lower bound $\widehat{\alpha}(I)\geq a/b$ with $a=\alpha(I)\operatorname{ht}(I)$ and $b=\operatorname{ht}(I)$, yielding $\widehat{\alpha}(I)\geq\alpha(I)$. Together with the general inequality $\widehat{\alpha}(I)\leq\alpha(I)$, this would prove equality for ideals with the packing property.