Geometric filtration conjecture for principal series

Let $\widetilde U=\widetilde U(\mathfrak g):=U(\mathfrak g)\otimes_{{\mathcal Z}(\mathfrak g)}S(\mathfrak h)$, and let

$$Pr:{\mathcal M}(\mathfrak t,M)\to {\mathcal M}(\widetilde U,K),\qquad V_w:{\mathcal M}(\mathfrak t,M)\to {\mathcal M}(\widetilde U,MN),$$

with ${\mathcal C}:{\mathcal M}(\widetilde U,K)\to {\mathcal M}(\widetilde U,MN)$. Geometric filtration conjecture. The functor

$${\mathcal C}\circ Pr:{\mathcal M}(\mathfrak t,M)\to {\mathcal M}(\widetilde U,MN)$$

admits a canonical $W$-filtration whose $w$-th subquotient is isomorphic to $V_w$. The source presents this as a more general formulation of the preceding geometric filtration claim and gives no resolution.