Converting Fire to Chaos question

I want to convert Blackwater Cocktail’s Fire Damage to Chaos Damage (probably with Mythical Sigil of Ch’thon if I remember the name correctly). And I thought that it is a little weird situation, because BC has burn damage, so Fire DoT, while Chaos doesn’t have DoT. After the conversion, will it be Chaos+Burn? If I would convert it to, let’s say, Cold, then I would have Cold+Frostburn, right? So what with Chaos?
Because I’m pretty sure there is no “Chaosburn” or anything like that.

Yes, it will go Chaos and the burn stay still

Thanks!
This will be perfect for the character I’m making :smile: