Chat GPTはgerundだと主張するがgerundではない。
present participleだという考えがある。
出典：
"That girl keeps smiling at him!"
"Smiling at him" is a participial clause, but it's not functioning as an adjective, and nor is "keep" a so-called linking verb.
This is a catenative construction in which "keep" is a catenative verb and the participial clause is its catenative complement. Within the participial clause, the preposition phrase "at him" is complement of "smiling".
It makes no difference whether the verb is just "keep" or the two-word idiom "keep on".