The symmetric Theta Conjecture for rooted trees with a prescribed root label

Let $n$ be a positive integer, let $\mathsf{RTT}_j(1^{n+1})$ denote the set of standard fully tiered rooted trees on $n+1$ vertices whose root is labelled $j$, and let $\mathsf{inv}(T)$ be their inversion statistic. Symmetric \Theta Conjecture. For $1\leq j\leq n$,

$$\left. \left\langle \Theta_{e_1^{j-1}} \Delta_{e_1} \Theta_{e_1^{n-j}} e_1,e_1^n \right\rangle \right\rvert_{t=1}=\sum_{T\in\mathsf{RTT}_j(1^{n+1})}q^{\mathsf{inv}(T)}.$$

This refines the preceding Theta Conjecture by separating the contribution according to the label of the root. The source motivates it using a decomposition of $\Theta_{e_1^n}e_1$ and gives no proof; the conjecture remains open.