Ok I can confirm my suspicion, having 2 or more abilities on cooldown will have to wait for all cooldowns to complete.
Taking a look at the Lyra project it is seems that each ability will have to have it’s own unique cooldown
Reference:
for example, GA_Grenade will have GE_Grenade_Cooldown
I think that there is a bug where the instigator of the effect is actually the player and not the ability, which means the caller is the player (or who ever called the ability)
It also might explain why do we have to grant (to the target) a cooldown tag in order for the ability have a cooldown, that is because in this case the target is also the instigator.