A conjectural divided-difference formula for products of operators

Let $i$ and $n$ be positive integers, let $z_j$ be variables, let $\mathcal{D}_j$ be the operators appearing in the modified Schubert-polynomial formalism, and let $\partial_j$ denote the corresponding divided-difference operators. Conjectural operator identity. The following holds:

$$\mathcal{D}_i (\mathcal{D}_{i+1} \mathcal{D}_i) \cdots (\mathcal{D}_{i+n-1} \mathcal{D}_{i+n-2} \cdots \mathcal{D}_i) = \prod_{j=i+1}^{n+i} \frac{z_j^{j-i}}{z_{n+2-j}^{j-i}} \partial_i (\partial_{i+1} \partial_i) \cdots (\partial_{i+n-1} \partial_{i+n-2} \cdots \partial_i) \prod_{j=i+1}^{n+i} z_{n+2-j}^{j-i}.$$

The source presents this as a conjectural compact formula relating products of the modified operators to divided-difference operators; its resolution and any combinatorial or Yang–Baxter-based proof are not specified in the supplied text.