I’m not sure of the rule on that. I would use or. That would be the most common way of writing it.
The big debate is whether none is plural or singular. In American English, the vast majority of people would use it in the plural. So it would be none were.
Scholars will argue about this. They will get into etymology and go back hundreds of years. They’ll declare, “My favorite English king was better than your favorite English king, and this is the way he wrote it. Therefore I’m right.”
I just know that in American English, it’s usually plural. This might be a case of differences between American and British English. I know words like “band” or “team” are singular in American English, and plural in British English.
The team is
The team are.
(Or so I’ve heard)
This might be true of none also.