Let $G$ be a (connected) reductive group over some ground field $F$ and $G^\*$ its unique quasi-split inner form. Denote by $\operatorname{rank}\_F G$ the split rank of $G$, i.e. the dimension of a maximal $F$-split torus in $G$, and likewise for $G^\*$. Is it true that 

$$\operatorname{rank}\_F G\le \operatorname{rank}\_FG^\*$$ 

with equality holds only if $G$ is isomorphic to $G^\*$? 

For example, this will say that among unitary group that splits over a fixed separable quadratic extension $E/F$, the quasi-split one has the largest split rank (which is $\lfloor\frac{n}{2}\rfloor$ for $U\_n$), and is the only unitary group that achieves this split rank. For split groups this result will be obvious.

I am mostly wondering about the case when $F$ is a non-archimedean local field (for Local Langlands, where the same Langlands parameter can be attached representations of both $G$ and $G^\*$), but any suggestion/reference will be great. Thanks!