The Tamari-interval conjecture for inverse images of non-crossing trees

Let $T$ be a non-crossing tree of degree $n$, not necessarily based, viewed as an element of the operad of moulds $obreak\operatorname{Mould}$. Let $\operatorname{Dend}$ denote the dendriform operad, and let $\mathcal{Y}(n)$ be the Tamari poset indexing a basis of $\operatorname{Dend}(n)$. Tamari-interval conjecture. The inverse image of $T$ in $\operatorname{Dend}$ is

$$\sum_{t\in I}t,$$

where $I$ is some interval in the Tamari poset $\mathcal{Y}(n)$. This conjecture has been checked in low degrees and predicts a precise Tamari-poset structure for these inverse images; the supplied text does not indicate a general proof or disproof.