Shoni are putting a tough resistance at Bungo, only province I haven't conquered from Kyushu. They now have a very strong army there.
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How did you get on with this situation?
Are you in a position to attack the province from two fronts now?
If so lure their army away from the castle, (tricking them into attacking a small insignificant, force and then retreating can work)
After luring them away you can take the province with a secondary force and then defend the province from a siege point of view which can be much easier..
Also if you time it and take the province at winters end turn they will be left in your newly acquired province and suffer winter attrition which can help greatly if numbers are tight..