BFP conjecture on nut graphs with two vertex and three edge orbits

Let $n$ be the order of a graph, and let a nut graph be a graph whose adjacency matrix is singular with one-dimensional null space spanned by an eigenvector having no zero entries. Vertex and edge orbits are the orbits of the automorphism group on the vertex set and edge set, respectively. BFP's conjecture. For every non-prime integer $n\geq 9$, there exists a nut graph $\Gamma$ of order $n$ with two vertex orbits and three edge orbits. The conjecture refines the known existence of nut graphs with two vertex orbits for every non-prime order $n\geq 9$ by requiring exactly three edge orbits.