Existence of nut graphs with two vertex orbits and three edge orbits

Let $n\geq 9$ be a composite integer. A nut graph is a graph whose adjacency matrix has nullity one and whose nullspace is spanned by a vector with no zero entries; write $o_v(G)$ and $o_e(G)$ for the numbers of vertex and edge orbits of $G$, respectively. The preceding theorem establishes existence with $o_v(G)=2$ for every such $n$. Existence conjecture. For every $n\geq 9$ that is not prime, there exists a nut graph $G$ of order $n$ with

$$o_v(G)=2\qquad\text{and}\qquad o_e(G)=3.$$

The conjecture strengthens the established two-vertex-orbit existence result by requiring exactly three edge orbits; the source gives no resolution beyond describing this as a belief about the corresponding table row.