Rubey's lattice conjecture for generalized chute moves

For a permutation $w$, let $\rp(w)$ denote its set of reduced pipe dreams, and order it by generalized chute moves. Rubey's lattice conjecture. This poset is a lattice. Certain special cases are known, including dominant, vexillary, and $1432$-avoiding permutations, but the conjecture remains open in general.