Explicit Schubert-product map commuting with the word-deletion operator

Let $\mathcal M$ and $\mathcal{RW}$ be the colored shuffled-word and reduced-word sets, and let $\psi:\mathcal M\to\mathcal{RW}$ be a map whose linear extension makes the diagram in Proposition commute. Let $\xi$ be the operator on words that removes the first letter.

Commuting-map conjecture. There exists a map $\psi$ satisfying Proposition such that

$$\psi\\,\xi=\xi\\,\psi.$$

Such a map would provide a combinatorial realization of Schubert structure-constant positivity; the conjecture strengthens the required properties of the map from Proposition.