Conjecture on minimal-expression independence of narrow screening operators

Let $\Lambda$ be the weight lattice and $W$ the Weyl group. For $\lambda\in\Lambda$, choose a minimal expression $\sigma=\sigma_{i_n}\cdots\sigma_{i_1}$ in simple reflections, and let

$$F_{\sigma,\lambda}\in\operatorname{Hom}_{\mathbb{C}}\bigl(V_{\sqrt{p}Q+\lambda},V_{\sqrt{p}Q+\sigma\ast\lambda}\bigr)$$

be the resulting composition of narrow screening operators. Let $\Phi_\lambda$ denote the map into the ambient screening realization.

Narrow-screening conjecture. For every $\lambda\in\Lambda$ and $\sigma\in W$, $F_{\sigma,\lambda}$ is independent of the choice of a minimal expression of $\sigma$. Moreover,

$$\Phi_{\lambda}(H^0(\xi_\lambda))=\bigcap\operatorname{Im}F_{\tau,\tau^{-1}\ast\lambda},$$

where the intersection is over all $\tau\in W$ for which $F_{\tau,\tau^{-1}\ast\lambda}\neq 0$.

This conjecture asserts both braid-word independence for the composed screening operators and an intrinsic description of the image of $H^0(\xi_\lambda)$ as an intersection of screening images. No resolution or partial result is supplied in the given text.