Yakimov's containment conjecture for quantum partial flag varieties

Let $G$ be a split, simply connected, semisimple algebraic group over a field $K$ of characteristic zero, let $P_I$ be the standard parabolic subgroup associated with a set of simple roots $I$, and let $W$ be the Weyl group of $G$. Write $W^I$ for the minimal-length representatives of $W/W_I$, where $W_I$ is the parabolic subgroup corresponding to $P_I$, and set

$$S_{W,I}:=\{(w,v)\in W^I\times W\mid v\leq w\}.$$

The ${\mathcal H}$-invariant prime ideals of $R_q[G/P_I]$ are parameterised by $S_{W,I}$; denote the corresponding prime by $P_{w,v}$. Yakimov's conjecture. For $(w,v),(w',v')\in S_{W,I}$, the containment $P_{w,v}\subseteq P_{w',v'}$ holds if and only if there exists $z\in W_I$ such that

$$w\geq w'z\qquad\text{and}\qquad v\leq v'z.$$

Yakimov states that this is proved in the full flag case but remains open in general, including the Grassmannian case; the supplied text does not report a resolution here.