"A wildy out of parameter rear could do what you say: chain toss when you're on both outer cog and outer chain ring.
In my opinion, That's possibly indicative of both chain stays bent to the R. They may be parallell, but over to the R."

Actually, if anything, they seem possibly bent to the left. Don't believe the problem is with the fork or the head tube. The string test indicates no error for the head tube/seat tube relationship and a lateral view of the forks and their relationship to the head tube seems okay.
Will have to photograph it.