Positivity of products with Lascoux polynomials

Let $\mathbb N^n$ denote the set of compositions indexing the Lascoux polynomials $\mathfrak L_\alpha$, and let $i\in[n]$. Lascoux product-positivity conjecture. For every $\alpha\in\mathbb N^n$ and $i\in[n]$, the product

$$x_1\dots x_i(1-x_{i+1})\dots(1-x_n)\mathfrak L_\alpha$$

is a graded nonnegative linear combination of Lascoux polynomials. This conjecture is of independent interest and is stated as a sufficient condition for the preceding general Lascoux-positivity conjecture; the supplied text gives no resolution.