You can't use a skill challenge in this situation. Apart from anything else, in the situation you're describing there are players on both sides of the conflict - whereas a skill challenge does not have room for mechanically active opposition. (You can run a skill challenge where the players/PCs are trying to achieve different things - the DMG2 gives an example of how to do this - but the opposition has to be mediated via NPCs upon whom the PCs operate directly.)
The difference is that the NPC doesn't get to make rolls - skill challenges are player-rolled skill checks against fixed DCs - and the NPC's behaviour, backstory etc can be metagamed in order to support the skill challenge resolution.
But even in a system that does, mechanically, permit player-vs-player - eg the Duel of Wits in BW - the losing player isn't forced to change his/her PC's mind. S/he just has to find some way to attack the outcome of the Duel in a collateral fashion (direct reopening of the debate being forbidden by the rules). In the context of BW, I see this as a particular application of Let it Ride.
Whereas the mental and social combat rules in Strands of Fate do require the affected characters (PC or otherwise) to alter their outlook. Potentially all characters can be affected as much as physical combat can affect all characters as well.