Partial flag Schubert polynomial limit conjecture

Let $(\mathbf n,\mathbf m)$ be as above, and let $\boldsymbol\alpha_{\mathbf n}$ and $\boldsymbol\beta_{\mathbf m}$ be positive sequences satisfying the stated conditions and summability assumptions. Suppose $u^{(n)}\in X_{P_{(\mathbf n,\mathbf m)}^{(n)}}(\mathbb R_{>0})$ converges uniformly to $u^{\infty}$ with

$$p_{u^{\infty}}(x)=\frac{\prod_{\beta\in\boldsymbol\beta_{\mathbf m}}(1+\beta x)}{\prod_{\alpha\in\boldsymbol\alpha_{\mathbf n}}(1-\alpha x)}.$$

For $w\in W^{P_{\mathbf m}}$, let $\widehat w_n=w_0^{(n)}w(w_0^{(n)})^{-1}$, and let $S_w$ denote the Schubert polynomial. Partial flag Schubert polynomial limit conjecture. The limits are

$$\lim_{n\to\infty}\mathfrak S^{\widehat w_n}_{(n)}(u^{(n)})=S_w\left(\frac1{\beta_1},\frac1{\beta_2},\frac1{\beta_3},\dots\right),$$

and, for $w\in W^{P_{\mathbf n}}$,

$$\lim_{n\to\infty}\mathfrak S^{w}_{(n)}(u^{(n)})=S_w\left(\frac1{\alpha_1},\frac1{\alpha_2},\frac1{\alpha_3},\dots\right).$$

These formulas describe the expected stable limits of Schubert functions on arbitrary partial flag varieties; the source provides no resolution evidence.