Conjecture on regular nut graphs from circulant and Cartesian graph constructions

Let $\mathcal{D}(n,t)$ denote the graph used in the construction, let $F_3$ and $F_5$ be the graphs shown in the paper, and let $\operatorname{\Box}$ denote the Cartesian product of graphs. A nut graph is a graph whose adjacency matrix has nullity one and whose corresponding nullvector has no zero entries. The constructions below are considered for integers $t\geq 1$. Regular nut graph construction conjecture. The following statements hold:

1. $\mathcal{D}(4t+6,t)\operatorname{\Box}F_5$ is a nut graph of degree $4t+5$ for every $t\geq 1$.
2. $\mathcal{D}(4t+6,t)\operatorname{\Box}F_3$ is a nut graph of degree $4t+3$ for $t\geq 1$ such that $t\not\equiv 0\pmod{3}$.
3. $\mathcal{D}(4t+10,t)\operatorname{\Box}F_3$ is a nut graph of degree $4t+3$ for $t\geq 1$ such that $t\equiv 0\pmod{3}$.

These constructions would provide infinite families of regular nut graphs in the specified degrees. The claim is presented as a conjecture in the source, and no resolution is supplied here.