The boost skills have to proc on attack like any other skill. However GW is an arena. Meaning Gluck will use speed order and not slow/atk boost female. Same with Schrif, Celestina, Demetria, Sheilah, Verlegen, Aqua, Brigitte, Entia, Fiato, Mentira, Rin, Rossetto, it seems like every unit with a split skill atk boost + something has an arena skill as well which will be used instead. Units that have atk boost that isnt split with another skill however seem to not have arena skills such as Camille, and Alex.
Split skill will still be used in colosseum battles though people have said only 1 skill or the other will proc if they get a skill activation. Not both and the atk boost again must activate and seems to go away after the effected unit attacks by my observations.