Zariski-dense Hitchin representations in uniform lattices

We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\mathrm{SL}(n,\mathbb{R})$, $\mathrm{Sp}(2n,\mathbb{R})$, $\mathrm{SO}(k+1,k)$, and ${\mathrm{G}}_2$. These subgroups are images of Hitchin representations. In particular, we show that every uniform lattice of $\mathrm{Sp}(2n,\mathbb{R})$, of $\mathrm{SO}(k+1,k)$ with $k\equiv 1,2[4]$ and of ${\mathrm{G}}_2$ contains infinitely many mapping class group orbits of Zariski-dense Hitchin representations of fixed genus. Together with Long-Thistlethwaite and with a previous paper of the author, it implies that all lattices of $\mathrm{Sp}(4,\mathbb{R})$ contain a Zariski-dense surface subgroup.