Use of "often" as antecedent

I understand that "often" can be used before a participle adjective and takes in that case a hyphen (e.g. "often-cited remarks" or "often-damaging winds").

Does it take a hyphen if it is followed by a regular (non-participle) adjective (e.g. "The company made billions of dollars on their often fraudulent claims of government contracts." or "What place do these personal confessions have in this often formulaic discourse of abstract theology?")

Should these be "often-fraudulent" and "often-formulaic" or in this case, is a hyphen not necessary?