Connectivity of the reduced bumpless pipe dream graph

For each $n$, consider the set of reduced bumpless pipe dreams of size $n$. A $2\times2$ flip is a local replacement of a valid $2\times2$ tile block by another valid tiling with the same boundary conditions. Only flips that preserve reducedness are allowed.

Connectivity conjecture. For every $n$, the set of reduced bumpless pipe dreams of size $n$ is connected under the $2\times2$ flips that preserve reducedness.

The unrestricted flips connect the full ASM lattice, but reducedness may obstruct this property and causes the standard monotone coupling approach to fail. Establishing connectivity would justify using these local moves on the reduced state space, although the paper does not resolve the conjecture.