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 $\lfloor\frac{n-1}{2}\rfloor$ 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!