Positivity conjecture for pre-canonical bases in type A

Let $\Phi$ be a root system of type $A_n$, let $X^+$ denote the dominant weights, and let $\mathbf{N}^i$ be the pre-canonical bases of the spherical Hecke algebra. Positivity conjecture. For each $i\geq 0$ and each dominant weight $\lambda\in X^+$, one has

$$\mathbf{N}_{\lambda}^{i+1}\in\sum_{\mu\in X^+}\mathbb{N}[q]\,\mathbf{N}_{\mu}^{i}.$$

This is presented as the central conjecture of the paper; formulas are known in several low-rank cases and a simple formula is proved for $n/2+1\leq i\leq n$, but the asserted positivity for all $i$ remains open.