Kimmerle's prime graph conjecture

Let $G$ be a finite group. Write $\#(G)$ for the set of primes dividing the order of $G$. The Gruenberg–Kegel graph, or prime graph, of $G$ is the graph $\pi(G)$ whose vertices are the primes in $\#(G)$, with an edge between distinct primes $p$ and $q$ when $G$ contains an element of order $pq$. Define $\pi(V(\mathbb ZG))$ analogously for the normalized unit group $V(\mathbb ZG)$. Kimmerle's conjecture. If $G$ is a finite group, then

$$\pi(G)=\pi(V(\mathbb ZG)).$$

This is a weakened variation of the Zassenhaus conjecture concerning the prime divisors of element orders in finite groups and their normalized integral group-ring units. It was proposed as a related approach to the Zassenhaus problem and has been studied for numerous classes of finite groups.