Greatest-element conjecture for permutations of a ribbon partition

Fix a partition $\lambda$, and consider the ribbons whose row lengths are a permutation of the parts of $\lambda$, ordered by $G$ positivity: $A\leq B$ when $a_{A,\nu}\leq a_{B,\nu}$ for every partition $\nu$. Permutation-class extremal-element conjecture. The set of ribbons which are permutations of $\lambda$ has both a least element and a greatest element. The source presents this as a conjecture about the induced $G$-positivity poset and gives no resolution.