Rule reminder
"Pur" + gerund (present or past) expresses a concessive meaning equivalent to "even though / while / despite" — but ONLY when both clauses share the same subject. Example: "Pur essendo stanco, ha lavorato tutto il giorno." (Even though he was tired, he worked all day.) The gerund agrees in time with the main verb.
_____ stanco, ha continuato a lavorare fino a tardi.