Abstract
We prove that if $\Sigma$ is a closed surface of genus at least $3$ and $G$ is a split real semisimple Lie group of rank of at least $3$ acting faithfully by isometries on a Riemannian symmetric space $N$, then there exists a Hitchin representation $\rho\colon \pi_1(\Sigma)\to G$ and a $\rho$-equivarient unstable minimal map from the universal cover of $\Sigma$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$, $n\ge 4$, this disproves the Labourie Conjecture
