Conjecture on regular prime graphs of finite groups

Let $G$ be a finite group, let $\rho(G)$ denote the set of primes dividing character degrees of $G$, and let $\Delta(G)$ be the prime graph of $G$, whose vertices are the elements of $\rho(G)$, with two distinct primes adjacent when their product divides an irreducible character degree of $G$. Let $k\geq 2$ be an integer and suppose that $\Delta(G)$ is $k$-regular. Regular prime graph conjecture. (1) If $k\geq 5$ is odd, then $\Delta(G)$ is a complete graph of order $k+1$. (2) If $k\geq 4$ is even, then $\Delta(G)$ is either a complete graph of order $k+1$ or a $k$-regular graph of order $k+2$. (3) If $|\rho(G)|=k+2$, then $G$ is solvable. The theorem for $k=3$, together with the known classifications for $k\leq 2$ and the constructions of even-degree examples given in the paper, motivates this conjecture; the assertions for general $k$ remain open in the source.