Amara action skill with urad

I’ve made an amara build using the new action skill augment for her that increases cooldown rate with action skill kills. The gun I am currently using on the build has the urad anoint. I am using the frozen heart shield with ASA anoint and ties that bind with the cryo element. Also have an icebreaker deathless on. When I am close enough to freeze enemies with the cryo nova when I activate ties, they seem to die almost immediately from the bonus radiation damage. But this does not appear to activate the cool down skill, whereas if I’m far enough away that frozen heart doesn’t hit them and I have to shoot them a few times to freeze them, it activates the skill and my action skill is back up in like 2 seconds. Is this because of the urad anoint killing them? Or is it the frozen heart? Any thoughts would be appreciated.