Minimal-degree conjecture for intervals in the Tamari lattice

Let $\operatorname{\mathbf{Tam}}_n$ be the Tamari lattice, and let $\mathbb{D}_n$ be its refined interval polynomial in variables $x,y,\overline{x},\overline{y}$. An interval has degree $n-1$ in $(x,y,\overline{y},\overline{x})$ when the corresponding monomial has total degree $n-1$ in these four variables; call $(S,S)$ a simple interval. Minimal-degree conjecture. The only intervals of degree $n-1$ in $(x,y,\overline{y},\overline{x})$ are the simple intervals $(S,S)$. The preceding proposition establishes the lower bound $n-1$ for the degree in $(x,y,\overline{y})$, while the source gives no resolution of this four-variable claim.