The Shapovalov-preserving Weyl group representation conjecture

Let $U$ be a finite-dimensional $\mathfrak g$-module, where $\mathfrak g$ is a simple Lie algebra with Weyl group $W$ generated by the simple reflections $s_i$. For each simple root, let $e_i,f_i,h_i$ be the corresponding $\mathfrak{sl}_2$ generators, and define $\tilde s_i$ on every $\mathfrak{sl}_2$-string by

$$\tilde s_i f_i^k v=\frac{k!}{(n-k)!}f_i^{n-k}v,$$

where $e_i v=0$ and $h_i v=nv$.

Shapovalov-preserving Weyl group representation conjecture. There exists a representation of the Weyl group $W$ in $U$ such that each generator $s_i$ is mapped to $\tilde s_i$. The operators $\tilde s_i$ preserve the Shapovalov form, but the paper presents the existence of a Weyl group representation with these generators as a conjectural modification of the standard Weyl group action.