The prime graph odd-prime labeling conjecture

Let $G$ be a graph. A graph is prime if it admits a prime labeling, and odd prime if its vertices can be injectively labeled by $1,3,\ldots,2|V(G)|-1$ so that adjacent vertices receive relatively prime labels. Prime graph odd-prime conjecture. Every prime graph is odd prime. No graph known to the authors is prime without being odd prime, while many graphs are known to be odd prime without being prime; the conjecture was previously introduced in the cited literature and remains open.