Mount & Blade: Will the AI capture their last city? Or must I capture it myself?

Here's my situation. Please advise if you know how the AI works.

Everyone has beaten Swadia into their last city, Uxkhal. There are 12 Swadian lords inside, each with about 20 troops, plus 60 garrison. That's too many dudes for my modest Nordic crew, so I need one of the other factions' help.

But none of them can finish the job. Khergit Khanate tried for a bit, then left. Rhodoks tried, then left. And now my own Nord king is trying. I just want one of those morons to finally capture Uxkhal, because my army needs to be out there beating down the Khanate, not sitting on top of Uxkhal and molesting the fleeing lords.

So, can I leave my this (often lonely) siege and trust them to finish it? Or do I need to capture this city myself?