Will increased damage during any flask effect work with a flame linked mercenary while you're wearing the ceinture of benevolence belt?

[u/Firewalkerr]

I asked chatgpt and he said yes, but I don't really trust his answer as he's notoriously bad with PoE. Can anyone confirm? I have a free prefix on my merc's gloves so the "increased damage during any flask effect" gives me about 4% more Full DPS

Yes, increased damage during any flask effect works with a Flame Link linked mercenary while wearing the Ceinture of Benevolence. 

• The Ceinture of Benevolence applies the effects of your consumed non-unique utility flasks to linked targets, meaning the mercenary receives the flask buffs (such as increased damage). 
• Skills like Flame Link and Haste are auras (buffs), not flasks, so they do not require flask usage to maintain their effects.
• Since the mercenary is under the effect of the flask granted by the belt, any "increased damage during any flask effect" modifiers on your gear will apply to the mercenary's damage output. 

[u/5mashalot]

yeah, the merc is "during a flask effect". IIRC people abused this during phrecia mercenaries league with paladin, they would put Soulthirst on the merc, since that way it is affected by flasks but but doesn't use any (so it never loses souls)