Fundamental non-symmetric shift operator conjecture

Let $W$ be a Weyl group, let $\rho_k$ denote the associated weighted half-sum, let $l$ be the shift parameter, and let $S(\mathfrak{h})$ be the symmetric algebra of the Cartan space. For a linear character $\varepsilon$ of $W$, write $\mathcal{G}^{(\varepsilon)}_\pm(k)$ for a non-symmetric shift operator and $T_p(k)$ for multiplication by $p\in S(\mathfrak{h})$. Fundamental non-symmetric shift operator conjecture. For any linear character $\varepsilon$ of $W$, there exists a non-symmetric shift operator $\tilde{\mathcal{G}}^{(\varepsilon)}_\pm(k)$ with shift $\pm l$ such that every non-symmetric shift operator with shift $\pm l$ is of the form

$$\tilde{\mathcal{G}}^{(\varepsilon)}_\pm(k)T_p(k)\qquad\text{for some }p\in S(\mathfrak{h}).$$

In particular,

$$\mathcal{G}^{(\varepsilon)}_\pm(k)=\tilde{\mathcal{G}}^{(\varepsilon)}_\pm(k)\prod_{\alpha\in R^0_+,\,l(\alpha)=0}\left(T_{\alpha^\vee}(k)-k^0(\alpha)\right).$$

For the sign character, the product is absent, so the conjecture asserts that $\mathcal{G}^{(\varepsilon)}_\pm(k)$ itself is fundamental. The conjecture concerns the expected minimal-order factorization of non-symmetric shift operators, but no resolution is supplied in the source.