Ultimately, this was resolved in the Geoff Johns-headed Batman: Three Jokers miniseries that ran in 2020. Within the story, Batman does discover that there are three jokers, the Criminal (closest aligning to his Golden Age portrayal where he was seeking to gain power and money through criminal enterprises), the Clown (closest aligning to the more wacky Silver Age appearances where his focus was theatrics and spectacle), and the Comedian (Modern Age psychopath who seeks to spread pain and suffering through his cruel jokes) with the revelation that one is the original Joker and the others were created through use of the Joker chemicals with the three intending to create a fourth, using Joe Chill as the subject.
By the end of the storyline, two of the Jokers are killed off, with Batman believing that the original Joker was the survivor.
Batman escorts the arrested Comedian (the real Joker) to Arkham Asylum. Jason tells Barbara that he would like to be more than friends with her, but she rejects him as she cannot get past the blood on Jason's hands. The Joker reveals that he is aware of Batman, Batgirl, and Red Hood's secret identities, and claims he does not want what the other two desired. The Clown just wanted to see people suffer which the Joker finds mundane, and The Criminal was delusional since the idea of creating a Joker with an identity is pointless; in his own view, the Joker is the personification of mystery and chaos. The Joker reveals he convinced the other two that Joe Chill would be the perfect Joker, because he understood he would never be able to commit a crime more tragic than what Chill did to Bruce, and he alone wants to be Bruce's greatest pain until the day they die together, which he accomplished when he forgave Chill.
Jason writes a letter for Barbara and tapes it to her apartment's front door. Inside the letter, he confesses his love for her and is ready to abandon the Red Hood identity for good, if it means having a chance to be with her. Barbara never reads the letter, as it falls from the door and is collected by a janitor. Since Jason notes in the letter that he also gives Barbara a chance to pretend this never happened, he is left unaware of this. In the aftermath, Bruce comforts Chill at his deathbed. Bruce reveals to Alfred that he's known the Joker's true name all along, discovering it one week after their first encounter. It is also revealed that the Joker's pregnant wife Jeannie was not actually killed, but taken to Alaska, where she now lives with her son as part of a witness protection program. Bruce explains that the Joker's name must never be known, because if the world ever found out that he had a family, it would be national news and they would be targeted, either by the Joker himself or by someone seeking vengeance against the criminal.
It's worth noting that the canonicity of the series has been debated, having been released under the DC Black Label, and contradicting some aspects of The New 52 and DC: Rebirth, but that Johns has stated that he believes the story remains within continuity.