In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4):

It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ has type $G_2$ and $K$ is infinite.

Here $H$ is the maximal split torus of $G_2$, $\sigma$ is an automorphism of $K$ such that $\sigma^2\varphi=1$, $\varphi$ is the Frobenius, $U^1$ and $V^1$ are the subgroups of upper and lower unitriangular matrices stable under the exceptional automorphism of $G_2$ induced by $\sigma$ and the root length changing symmetry of the Dynkin diagram, the subgroup $G^1=\langle U^1, V^1\rangle$ and $H^1=H\cap G^1$.

My question: is this still the case?