Bumpless pipe dream transition compatibility

Let $B$ be a $k$-row reduced bumpless pipe dream such that row $k$ has no blanks (“weighty tiles”). Let $\phi(B)$ be the RC graph corresponding to $B$ under the natural bijection between bumpless pipe dreams and RC graphs, let $\mathbf{t}$ be Weigandt's transition map on bumpless pipe dreams, and let $\mathcal{Z}$ denote the RC-graph transition operation. Bumpless pipe dream transition conjecture. Then

$$\mathcal{Z}(\phi(B))=\phi(\mathbf{t}(B)).$$

This is the precise compatibility assertion behind the proposed realization of the RC-graph transition algorithm by bumpless pipe dreams. The source describes the transition maps and the Gao–Huang bijection, but gives no resolution of the conjecture.