There are two types of relation between the adjectival clause and the noun in adjectival modifications.
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2)彼が死んだといううそ: the lie that he dead
As I wrote in my previous post, the clause 彼が死んだ is the explanation regarding the content of うそ. うそ is not the subject/object/target of 彼が死んだ.
A linguist 寺村秀夫 named these relations 内の関係(inside-relation), 外の関係(outside-relation) respectively. という is needed only for the case of outside-relation.