I had no shimmy during the first 300 hours (the life of the cheapy tires that came with the kit). However, after I put new Michelin tires on, I had a fore-aft shimmy at 23 knots. I did put lead in the forwardmost part of the main wheel fairings, to no avail. I do think balancing them is a good idea, though, as it will minimize pitching moments on the fairing mounts as the tires roll over bumps.
Shimmy modes are simply resonances (defined as some input frequency being equal to some natural vibration frequency). The natural frequency of something depends only on its spring constant (in the direction of interest) and mass. Adding energy absorption clearly will help (lower air pressure, or wood, which is really only designed as an energy absorber, not a stiffener), but there is no general solution which will work for all.
As for me, I simply do not taxi at 23 knots.