The Theta Conjecture for fully tiered rooted trees

Let $u$ be a composition, let $u( u)$ denote the partition obtained by rearranging its parts, and let $\mathsf{RTT}(\nu)$ be the set of fully tiered rooted trees associated with $u$, with $\mathsf{inv}(T)$ their inversion statistic and $x^T$ their monomial weight. \Theta Conjecture. For any composition $u$,

$$\left. \Theta_{e_{\nu(\nu)}} e_1 \right\rvert_{t=1} = \sum_{T \in \mathsf{RTT}(\nu)} q^{\mathsf{inv}(T)} x^T.$$

The conjecture would imply that the tree expansion on the right is independent of the order of the parts of $u$, and it has been checked by computer for $|\nu|\leq 7$. Equality of the Hilbert series is known, but the full symmetric-function identity remains open.