Unstable minimal surfaces in symmetric spaces of non-compact type

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 at least $3$ acting faithfully by
isometries on a symmetric space $N$, then there exists a Hitchin representation
$\rho:\pi_1(\Sigma)\to G$ and a $\rho$-equivariant 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 $G=\mathrm{PSL}(n,\mathbb{R})$, $n\geq 4$, this
disproves the Labourie conjecture.