On the KNS Conjecture in type $E$

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.