Parabolic aligned elements form a lattice quotient of parabolic weak order

Let $(W,S)$ be a finite Coxeter system, let $J\subseteq S$, and let $c\in W$ be a Coxeter element. An element of $W^J$ is $(W^J,c)$-aligned if, whenever $t_{\alpha}<t_{a\alpha+b\beta}<t_{\beta}$ in $\operatorname{Inv}(\mathbf{w_{\circ}^{J}(c)})$, where $\alpha,\beta\in\Phi^{+}$ and $a,b$ are positive integers, the condition $t_{a\alpha+b\beta}\in\operatorname{Cov}(w)$ implies $t_{\alpha}\in\operatorname{Inv}(w)$. Write $\operatorname{Align}(W^J,c)$ for the set of such elements, and let $\operatorname{Weak}(\operatorname{Align}(W^J,c))$ denote the weak-order poset on this set. Parabolic lattice conjecture. For any finite Coxeter system $(W,S)$, any $J\subseteq S$, and any Coxeter element $c\in W$, the poset

$$\operatorname{Weak}\bigl(\operatorname{Align}(W^{J},c)\bigr)$$

is a lattice. Moreover, it is a lattice quotient of $\operatorname{Weak}(W^{J})$. This would extend the known parabolic Tamari-lattice result from the symmetric group to arbitrary finite Coxeter systems and parabolic quotients; the claim was motivated by computer experiments, but no resolution is supplied here.