Conjecture on Green functions for Levi subgroups of type A

Let $\mathbf G$ be a group of type $A$, let $\mathbf L\leq\mathbf G$ be an $F$-stable Levi subgroup, and let $v\in\mathbf L_{\operatorname{uni}}^F$. Let $u$ be the induced class of $v^{\mathbf L}$ in the sense of Lusztig--Spaltenstein. Green-function conjecture. Then

$$\widetilde{Q}_{\mathbf L}^{\mathbf G}(u,v)=1.$$

The examples suggest that the modified 2-parameter Green function takes this value whenever $u$ is the induced class of $v^{\mathbf L}$ for groups of type $A$; the conjecture is stated based on these examples, and no resolution is given here.