Weight on the front wheel would have no effect on whether it's not 'true'......
I would rig a pencil held horizontally against the side of the rim fixed to a coat-hanger , sitting on the floor and bent so to support the pencil just touching the rim...spin wheel slowly..blah, blah you know what I mean, but also see does pencil 'ride' higher or lower on rim ( flat spot )....good luck....