Down-up Monk connectedness conjecture for back-stable Schubert products

Let $S_{\mathbb Z}$ be the group of finitely supported bijections of $\mathbb Z$. A down-up Monk move is a down-up Bruhat move $wt_{a,b}t_{c,d}=w'$ satisfying either $a\leq c<b\leq d$ or $c\leq a<d\leq b$. Equivalently, it connects two permutations occurring with nonzero coefficients in a product with a simple reflection. A set is down-up Monk connected if every two of its elements can be joined by a sequence of such moves.

Down-up Monk connectedness conjecture. For any $u,v\in S_{\mathbb Z}$, the set $\\{w\mid\overleftarrow{c_{u,v}^w}\ne0\\}$ is down-up Monk connected.

This strengthens down-up Bruhat connectedness by restricting the allowed moves according to Monk's rule; it remains open.