Bašić et al.'s existence conjecture for circulant nut graphs of degree divisible by four

A circulant nut graph is a circulant graph whose adjacency matrix has a one-dimensional null space spanned by a full vector. Let $d$ be a degree satisfying $d \equiv_4 0$, and let $n$ be even with $n \ge d+4$. Bašić et al.'s conjecture. For every such $d$ and $n$, there exists a circulant nut graph of order $n$ and degree $d$. The conjecture gives a sufficient existence condition matching the necessary condition for vertex-transitive nut graphs in the case that the degree is divisible by four; its resolution is not indicated in the supplied text.