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So, basically Crom thought this would make for a great show so he summons these two and forces them to fight.

They're in character but know the only way they're walking away from this is if one of them dies.

We're going by everything for both, and that includes the comic books. As well as the novels, and the video games.

Scenario 1) They have a sword fight.

Scenario 2) No holds barred they use everything at their disposal. Which means they have all their weapons, armor, and equipment they ever had. Along with any other abilities they picked up.

I have no idea what comic/book Geralt is capable of. But if the creatures he fights are any indication in the games(Wild Hunt or dragons or golems, etc...), then Conan gets one of the hardest fights he's ever had.

S1 - I want to say Geralt, but Conan has a Captain America-esque plot armor where he just wins.

S2 - Definitely Geralt. His Signs clinch this victory for him easy.