WI: Henry III of England Does Better in Trying to Conquer France

Henry III of England tried to reconquer the lost territories in France his father John I lost to the French. Unfortunately for him his attempts at retaking those lands ended up in failure every single time. However if Henry III does a better job and enjoys some better success (does not have to be regaining everything John I lost), how does this impact the rest of his reign? Does this improve England's ability to conquer the remainder of France? And how is France affected in general? How is Simon de Montfort's life affected?
 
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