Did Pudgalavada Buddhists claim the the "person" existed between lives and / or in one life?

Did Pudgalavada Buddhists claim the the "person" existed between lives and / or in one life? The Pudgalavadins were an old school of Buddhists who thought there was a person. I'm currently unclear whether that is ultimately, in the sense that Theravada and Mahayana Buddhists use the term 'ultimate'. They debated about this with e.g. the Vibhajyavada sect, which I think later became Theravada Buddhism. You can read about the canonical dismissal of it in Theravada Buddhism in the 1st chapter of their Kathavatthu . ---------- To clarify, one of my links say > It is this self, they maintained, that dies and is reborn through > successive lives in Samsara, continuing to exist until enlightenment > is attained. Is this the *same* person, each life?