Principality of Antioch

In 1149, Raymond of Poitiers, Prince of Antioch, was killed in battle with Nur ad Din of Aleppo. Suppose Antioch had fallen to the Zengids not long after the battle. Would a new expedition have come from the West? Or would this have given an opportunity for the Byzantines to get control of Antiochn?
 
The Second Crusade had just ended the year before in a very unsatisfactory manner for the Catholic Christians. Doubtful that the will power or the resources are available to launch another Crusade.

Any chance for a reconquest of Antioch would need to come from Constantinople.
 
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