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

Let $n$ be the order of a circulant graph, and let $\mathrm{Circ}(n,S)$ denote the circulant graph with generator set $S$. A nut graph is a non-trivial graph whose adjacency matrix has nullity one and whose non-zero null-space vectors have no zero elements. Bašić et al.'s conjecture. For every $d$ with $d\equiv 0\pmod 4$, and for every even $n$ with $n\geq d+4$, there exists a circulant nut graph $\mathrm{Circ}(n,\{a_1,a_2,a_3,\ldots,a_{d/2}\})$ of degree $d$. This is an existence conjecture for circulant nut graphs of every degree divisible by four; the supplied text gives no resolution status.