AHC: The medieval Jin dynasty defeat the Song to unify China


Gone Fishin'
Could the Jin dynasty, who in OTL conquered northern China from the northern Song dynasty, unify China by conquering southern China as well?

What would be required for this to happen?

Jin enhancements to leadership or luck? Song screw-ups?

The Jin fell to the Mongols a couple decades before the southern Song, both gained some lands at Jin expense. Is Jin southward expansion at southern Song expense, and perhaps the total conquest of the latter, the default course of events if somehow the Mongols did not coalesce into externally conquering hordes?

(or was centuries more of north south partition more likely? Or a southern Song reclamation of the north?) I forgot what happened in the "Chaos TL".