Action of the word-deletion operator on back-stable key polynomials

Let $\alpha$ be a composition, let $\eta(\alpha)=\\{i\mid\nexists j<i,\alpha_j=\alpha_i\\}$, and let $\delta_i$ be the composition that is $1$ at index $i$ and $0$ elsewhere. Composition addition and subtraction are componentwise. Let $\overleftarrow{\kappa}_\alpha$ denote the back-stable key polynomial, and let $\xi$ be the operator that removes the first letter of a word.

Key-polynomial operator conjecture. For any composition $\alpha$,

$$\xi(\overleftarrow{\kappa}_\alpha)=\sum_{i\in\eta(\alpha)}\overleftarrow{\kappa}_{\alpha-\delta_i}.$$

The formula predicts an explicit action of $\xi$ on the back-stable key basis. The supplied text gives no evidence of resolution, so it is recorded as open.