Hamiltonicity of intersections involving more than two primes

Determine whether the Hamiltonian property extends to intersections of prime multiple missing graphs associated with larger numbers of odd primes; specifically, establish whether such intersections are Hamiltonian for even $n>2$ and consequently possess Hamiltonian paths for odd $n$.

Background

The paper proves that each graph $G(p,n)$ is Hamiltonian for even $n>2$, and that the intersection $G(3,n)\cap G(5,n)$ has the same Hamiltonian property. It emphasizes that intersections involving more prime factors are substantially more complicated and that the available results do not determine whether Hamiltonicity persists in those larger intersections. The unresolved issue is directly connected to the Hamiltonian structure of finite near Goldbach graphs, which are finite intersections of prime multiple missing graphs.