Three components of shimmy?
I have some experience looking at failures of some structures used in rail vehicles. I am not an expert but listened to some and read a lot and will offer some food for thought.
In the structures we worked on there were three components to the issue.
1) excitation source (stay cool now) - in our case it was steel wheels with a flat spot. They are 36" diameter wheels and at around 45 mph the frequency of the flat spot hitting the rail was right about at the harmonic frequency of the structure. Trains often run at 45 mph and when they did this structure would vibrate and fatigue the structure eventually causing some cracks in some equipment mounted to the car.
2) structural harmonic frequency - structures have a frequency that they will most easily vibrate. Most structures are so stiff that the frequency is very high and there are few if any excitation sources at the same frequency. Sometimes structures are designed and built where mass, stiffness and inherent dampening provide a structure that is easily excited at a frequency commonly found.
3) dampening - Sometimes the material or construction methods will quickly absorb or dampen any vibration.
In the case of this problem I suspect:
1) there are many possible excitation sources and variables including wheel geometry (toe in and camber), wheel/tire balance, tire pressure, frictional coefficient between tire and surface, and probably others
2) structural harmonic frequency is an issue with some gears and not others because of structural stiffness and the way the mass of the landing gear, wheel/tire/axle/brake system and wheel pant is distributed. These gears are basically a long beam with a mass at the end. If you change the mass or it's position you will change the harmonic frequency. This would include adding stiffeners or maybe changing tires or wheels/brake discs. I would guess Grove gears and Vans gears (on 8's) have different natural frequencies due to slightly different shapes (therefore stiffness) and distribution of mass. Longer gears would be different than short, tapered tube different than a constant cross section gear etc. Even clamping force in the gear mount could effect the harmonics.
3) dampening - I don't see where there is any real dampening in this system. Everything is steel except the tires and I doubt they provide any dampening, only a poor connection to the ground.
Think of this landing gear as a tuning fork. If the fork does not get excited (no source) or the excitation frequency is a lot different than it's natural frequency, or if you lightly touch the vibrating tuning fork with your fingers the vibration will either not occur or will dampen out quickly.
The moral of the story is if you eliminate the source of the exitation source, or change the natural frequency of the structure (stiffeners) or add dampening you will probably eliminate or reduce the vibration, or shimmy in this case. That's why some people get it bad, some get it in one gear leg only and some people (airplanes) never exhibit it with enough energy for the pilot to notice it.