Originally Posted by Jipe

35x406 without mudguards (Continental Contact Urban 32x406 on 25mm inside rims = 35mm measured width).

Pacific Cycles deliver the R20 11SP with Panaracer Minits Tough 32x406.

I didn't had any folding issue but it was with the Titanium Birdy, not Birdy III. It might be different with a Birdy III but I doubt about it because the overall wheel size isn't much bigger in 35x406 than in 50x355 which is the standard wheel size of the Rohloff Birdy III.
Below a picture of the folded Ti Birdy with the 32x406 Continental Contact Urban tires (and a custom carbon front mudguard+Ridea rear mudguard and easy wheels):