In the context of the sentence, “the kind of hats that you knew they didn’t really live in New York,” the word "that" functions **solely as a complementizer, not as a relative pronoun**.

Here's why:

1. **Complementizer Role**: In FRCs, "that" serves as a bridge or connector between the head noun ("the kind of hats") and the modifying clause ("you knew they didn’t really live in New York"). Its role is to introduce the clause rather than act as a noun within it.

2. **Relative Pronoun Absence**: A relative pronoun would serve as a placeholder for the noun within the clause itself (e.g., "which" in "the hats which they wore"). In FRCs, however, "that" does not substitute for a noun or play an internal role within the modifying clause; it simply marks the start of the clause.

3. **Obligatory in FRCs**: The use of "that" in this sentence as a complementizer is syntactically required to maintain the structure of the FRC. In contrast, in restrictive RCs, "that" is often optional or can be replaced with a relative pronoun, depending on the clause’s internal structure and meaning.

Therefore, in this sentence, "that" is **only a complementizer** and does not function as a relative pronoun.

Yes, the author argues that in Frame Relative Clauses (FRCs), the word "that" operates exclusively as a **complementizer** rather than a **relative pronoun**. The distinction is central to their argument because it emphasizes how FRCs differ syntactically from traditional restrictive and non-restrictive relative clauses within Cinque’s framework.

The paper outlines several points supporting this view:

1. **Obligatory Presence of "That" in FRCs**: The author notes that in FRCs found in non-standard varieties of English (as in the sentence “the kind of hats that you knew they didn’t really live in New York”), the complementizer "that" is required, which differs from restrictive RCs where "that" can often be omitted. This obligatory presence is seen as a structural requirement unique to FRCs.

2. **No Relative Pronoun Function**: The author argues that "that" does not act as a relative pronoun (i.e., it does not serve as a syntactic placeholder for the noun within the clause). Unlike in typical RCs where relative pronouns like "who," "which," or "that" refer back to the head noun within the clause, "that" in FRCs does not represent any specific noun or play an internal role in the clause. Its role is simply to introduce the modifying clause.

3. **Distinction from Standard Relative Clauses**: By analyzing "that" as a complementizer rather than a relative pronoun, the author underscores the structural and functional uniqueness of FRCs. This helps to illustrate why FRCs do not align with the typical RC categories in Cinque's (2020) typology, such as restrictive or non-restrictive RCs.

In summary, the author's argument is that "that" in FRCs is strictly a complementizer, and this distinction is one of the syntactic features that separates FRCs from other relative clause types within Cinque’s unified framework.