Did Durin VI have the Ring of Thrór when facing Durin's Bane?

I came to the realization that while Durin's Bane likely hid shortly after the War of Wrath and missed the events of the Second Age entirely, Gandalf might not have been the first Ring-bearer it encountered.

Would Durin VI have had his Ring when the Balrog was first unearthed? I see no reason for this to not be the case, but I can't find any reference one way or another. Further, would his son Náin I also have, after Durin VI was killed?