Let $G$ be the base change of a connected reductive group over ${\mathbb{F}}_q$ to $\overline{\mathbb{F}}_q$, and let $F$ be the associated geometric Frobenius on $G$. Let $W$ be the Weyl group of a quasisplit $F$-stable maximal torus of $G$.

By definition, $F$ acts on $W$. We know that, the $G^F$-conjugacy classes of $F$-stable maximal tori in $G$ are one to one corresponding to the $F$-conjugacy classes of $W$, and the quasisplit ones form a single class corresponding to $1\in W$.

**Question:** Under this picture, is it true that, the $F$-stable maximal tori which are NOT contained in any $F$-stable parabolic subgroup of $G$ form a single class?

I checked standard references like Carter's book and Digne--Michel's book but couldn't find the answer; any reference containing a positive proof or counterexamples would be appreciated!