I have recorded video evidence! The battle goes like normal up until I summon Rogue One. Not only does it NOT reduce cooldowns on my ships but it does NOT reset the reinforcement cooldown on Profund...
Ideas
One last thing to note. I have proven to myself that it might be partly an interaction with Falcon that is dispelling the tenacity up. I swapped it out and did not have the same problem on the next battle. I can't verify any of this for sure even if it was recorded because only the coded abilities can say exactly what is happening.
Or maybe I need to keep testing it to verify.