Dual Barnette's Conjecture
Maybe I am mistaken but I think it is possible to make the example from https://math.stackexchange.com/a/78835/131224 Eulerian. Just take an octahedron and place 3 new vertices on each of its faces, connected to each other and also to two of the vertices of the face in a circular order. So if the face was $abc$, then add $def$ and connect $d$ to all but $a$, $e$ to all but $b$ and $f$ to all but $c$. The same reasoning works but now every degree is even.