  For the exceptional types $E_6$, $E_7$, and $E_8$, we prove that the
specializations at roots of unity of the quantum dimensions of the
Kirillov-Reshetikhin modules give real solutions of $\ell$-restricted
$Q$-systems, as conjectured by Kuniba, Nakanishi, and Suzuki. We also show that
these solutions are positive in type $E_6$. In type $E_7$ and $E_8$, we only
prove positivity for a subset of the nodes of the Dynkin diagram.
