On the Prime Graph Question for Integral Group Rings of Conway simple groups

The Prime Graph Question for integral group rings asks if it is true that if the normalized unit group of the integral group ring of a finite group $G$ contains an element of order $pq$, for some primes $p$ and $q$, also $G$ contains an element of that order. We answer this question for the three Conway sporadic simple groups after reducing it to a combinatorial question about Young tableaus and Littlewood-Richardson coefficients. This finishes work of V. Bovdi, A. Konovalov and S. Linton.