With Hexhead sizes in mind, this is going to sound strange given the blocky thread pattern......but you'll struggle to find a stickier tyre than the Avon Trailrider.
I'm going with similar advice too.
I never tried the 01, or any other 100% asphalt type of tyre, on the GS but I've used a number of pairs of Dunlop Trailsmart and I always found them excellent. The only bad thing I can say: the front tyre would get ruined towards end of life (little "steps" I don't remember what would be the term in English) but that could be due to suspension settings, pressure (I keep them a bit on the low side) and any riding errors on my side.
But again, excellent tyres. More than decent mileage and wear: