I submitted, as I didn't want to have to fight all of the guards and not be able to do other Markarth quests. Unfortunately, a 1k bounty remains in the log afterwards, even after being freed. Even if no one's trying to arrest me, I'd like to have a clean slate.
I didn't mind that, just reasoned it as a miscarriage of justice. Like when I killed a bunch of Thalmor agents who attacked me when I walked up to them and somehow recieved a 1000 gold bounty in Falkreath.
I had like 9 murders in my stats as well as 18 assaults, never killed or attacked an innocent in my entire playthrough. A bunch of Khajiit traders attacked me one time and I killed them all, counted as murder for some reason.
Same thing happened with Stormcloak soldiers in the CW questline and when attacking Stormcloak camps after I'd completed it.