The existence conjecture for 12-regular circulant nut graphs

A circulant nut graph $C(n,S)$ is a circulant graph on $n$ vertices whose adjacency matrix has nullity one and whose nullvector has no zero entries; here $S=\{a_1,a_2,\dots,a_6\}$ specifies the six positive circulant steps, giving degree $12$. The 12-regular circulant nut graph conjecture. For every even $n$, $n\ge 16$, there exists a circulant nut graph $C(n,\{a_1,a_2,\dots,a_6\})$ of degree $12$. The conjecture extends the explicitly constructed examples for several even orders between $16$ and $38$ and asks for such a graph at every even order at least $16$.