Nonexistence conjecture for cubic 8-circulant nut graphs

A cubic $8$-circulant nut graph is a cubic nut graph admitting an $8$-circulant structure. Nonexistence conjecture for cubic $8$-circulant nut graphs. There exists no cubic $8$-circulant nut graph. This is the only unresolved case in the cubic $\ell$-circulant nut graph existence problem; computational results suggest that such graphs might not exist, but no proof is given here.