So, I ran a test on this.
Two heroes: Avril and Gan
Avril does have level 2 Rapid Cast. Not sure if this affects the test, but in theory, it shouldn't.
Battle 1 (Resplendent Gear equipped): Ability cooldowns equal between Gan and Avril. Used Gan's ability. Avril abilities came up approximately 2 seconds before Gan's.
Battle 2 (Removed Resplendent gear): Exact same effect as battle 1.
So basically, nothing at all changed with or without the set. I wasn't sure if it was supposed to affect all hero cooldowns or only the hero wearing it, but it seemed to do nothing at all. Anyone else see this as true?