I think you mean "Relentless" gear, right?

However, if you read his A2 ability, it specifically says "This skill's cooldown cannot be decreased by other cooldown decreasing effects". That means ALL effects. So it doesn't matter if it's from another CD champ or gear, nothing can bring it down.

This has been the concensus of all the YT reviewers. I'm not sure if anyone has tried to test it but according to the wording, it shouldn't.

Double checked and no, nothing should reduce that cooldown. 

Is this typical coding?  Honestly never paid attention with other reset champs.

That is standard except for Yumeko.  

Plarium snuck in and added language to all of the other reset toons to clearly state that "this skill's cooldown cannot be reduced or reset."

They also gave Yumeko the opposite wording.

I have not tried to test Reflex gear on a Reset Toon, but I assume it will not work.  However, Relentless Gear would still do its thing because that is not cooldown reduction. That is extra turn.

oui. because void fox lady is special and everyone else isn't. :)