Motzkin-number conjecture for doubly minimal-degree Tamari intervals

Let $\operatorname{\mathbf{Tam}}_n$ be the Tamari lattice, and consider its intervals. For an interval, measure degree separately in the variable pairs $(x,y)$ and $(\overline{x},\overline{y})$. Motzkin-number conjecture. The number of intervals having degree $n-1$ in $(x,y)$ and degree $n-1$ in $(\overline{x},\overline{y})$ is a Motzkin number. The claim concerns a refined enumeration of Tamari intervals; the source gives no further specification of the indexing of the Motzkin number or evidence resolving the assertion.