  Let $G$ be a split real form of a complex simple adjoint group and let $S$ be
a closed orientable surface of genus at least 2. We show that for generic
$\operatorname{PSL}_{2k-1}(\mathbb{R})$-Hitchin representations $\rho$, the
middle eigenvalue of $\rho(x)$ is not equal to 1 for all non-trivial elements
$x\in \pi_1(S)$. Using the same technique, we prove that the set of strongly
dense $G$-Hitchin representations is generic. This generalizes the result of
Long, Reid and Wolff for the $G=\operatorname{PSL}_n(\mathbb{R})$ case.
