On cuspidal maximal tori of a connected reductive group

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!

No. For example, in (split, simply connected) type $\mathrm C_2$, there are the tori whose rational points are isomorphic to $\ker \mathrm N_{\mathbb F_{q^4}/\mathbb F_{q^2}}$, and those whose rational points are isomorphic to the product of $\ker \mathrm N_{\mathbb F_{q^2}/\mathbb F_q}$ with itself.
• I would like a reference discussing the possible $F$-stable maximal tori which are not contained in any $F$-stable proper parabolic subgroup. (A bit shame to say, but I don't familiar with the term elliptic torus'', does an $F$-stable elliptic torus never contained in $F$-stable proper parabolic?) – user148212 May 21 '17 at 14:25
• @user148212, the two tori that I've mentioned correspond, respectively, to the Coxeter element of $\mathrm C_2$, and to the Coxeter element of the $\mathrm A_1 + \mathrm A_1$ sub-system. – LSpice May 22 '17 at 2:51