Power-sum equality for orientation posets

Let $\mathbf{a}$ be an area sequence of length $n$, let $O(\mathbf{a})$ be the set of orientations associated with $\mathbf{a}$, let $P(\theta)$ be the poset obtained from the ascending edges of $\theta$, and let $B(\theta)$ be the disjoint union of chains whose lengths are given by $\overline{\pi}(\theta)$. For a partition $\lambda\vdash n$, let $\mathcal{O}^{\ast}_{\lambda}(P)$ denote the set of order-preserving surjections of type $\lambda$ whose fibers have unique minimal elements. Equivalent power-sum conjecture. For every area sequence $\mathbf{a}$ of length $n$ and every partition $\lambda\vdash n$,

$$\sum_{\theta\in O(\mathbf{a})}q^{\operatorname{asc}(\theta)}\left|\mathcal{O}^{\ast}_{\lambda}(P(\theta))\right|=\sum_{\theta\in O(\mathbf{a})}q^{\operatorname{asc}(\theta)}\left|\mathcal{O}^{\ast}_{\lambda}(B(\theta))\right|.$$

This is stated as an equivalent reformulation of the main LLT identity after comparing power-sum coefficients; the source gives no resolution status.