Cikakken jumla mai zuwa: Lokacin 2004/05 ya ga canji ga wannan manufar, yayin da ƙungiyar ta fara amfani da teburin Celtic League a matsayin ƙa'idodinta kawai don tantance waɗanne ƙungiyoyin Irish za a shigar da su a gasar Heineken Cup na kakar wasa mai zuwa. Connacht ya gama maki shida a bayan Ulster a cikin matsayi na karshe a wannan shekara, ya sake rasa shi. Ba sai
2016 cewa Connacht ya gama a matsayi mafi girma fiye da duk abokan hamayyarsu na lardin. Duk da wannan Connacht ya sami cancantar Heineken Cup daga kakar 2011-12 zuwa kakar 2013-14. Abin mamaki wannan ya kasance godiya ga Leinster, wanda ya lashe gasar Turai uku a jere. Nasarar Leinster na nufin cewa sun cancanci cancanta a shekara mai zuwa, suna barin buɗaɗɗen cancantar Irish wanda Connacht ya cika.