The even-parameter order-four circulant nut graph conjecture

Let $t$ be an integer and let $n$ be the order of a circulant graph. A circulant nut graph is a circulant graph whose adjacency matrix has nullity one and whose non-zero null-space vectors have no zero elements. The conjecture. For each even $t\geq 4$ and each $n\geq 4t+8$ divisible by four, there exists a $4t$-regular circulant nut graph of order $n$. The preceding results establish constructions for odd $t$ and for even $t$ when $n\equiv 2\pmod 4$, while the conjecture concerns the remaining divisible-by-four orders for even $t\geq 4$; the supplied text gives no resolution status.