The exceptional-collection conjecture for permutation representations of partial flag varieties

Let $G$ be an algebraic group, let $P\supset B$ be a parabolic subgroup, and let $\mathsf{k}[(G/P)(\mathbb{F}_q)]$ be the permutation representation over $\mathsf{k}$. Let $\widetilde{\rho}$ denote the virtual representation of $G$ constructed by the paper as a lift of a representation $\rho$ of $G(\mathbb{F}_q)$. Let $C_X^P=\{X_w\}_{w\in W}$ be the Samokhin–van der Kallen exceptional collection, and let $V_{C_X^P}$ be the associated virtual representation. The exceptional-collection conjecture. When

$$\rho=\mathsf{k}[(G/P)(\mathbb{F}_q)],$$

for some parabolic subgroup $P\supset B$, one has

$$\widetilde{\rho}=V_{C_X^P}.$$

The authors report computational verification in Types $A_2$, $B_2$, $G_2$, and $A_3$, but no general proof is given, so the conjecture remains open.