Unless stated otherwise, in these cases the conjecture is know to be induced by a `split endomorphism two-sided tilting complex'.
Thanks to Naoko Kunugi for help in preparing this page.
This general theorem includes, for p=3, some cases where G has Sylow 3-subgroup C3×C3, which were needed for the theorem of Koshitani and Kunugi. Namely: G=Sp4(q), where q=4 or 7 (mod 9); G=PSU4(q2) or PSU5(q2), where q=4 or 7 (mod 9) [see also Koshitani and Miyachi [pre-KosMiy]]. Of course, Puig's theorem also deals with the case q=1 (mod 9), but then the Sylow p-subgroup is larger than C3×C3.