Extract from Wikipedia
Clicky here:
Wobble or shimmy begins when some otherwise minor irregularity accelerates the wheel to one side. The restoring force is applied in phase with the progress of the irregularity, and the wheel turns to the other side where the process is repeated. If there is insufficient
damping in the steering the oscillation will increase until system failure. The oscillation frequency can be changed by changing the forward speed, making the bike stiffer or lighter, or
increasing the stiffness of the steering, of which the rider is a main component.
[2]While wobble or shimmy can be easily remedied by adjusting speed, position, or grip on the handlebar, it can be fatal if left uncontrolled.
[4]
Since shimmy frequency is independent of bike speed, gyroscopic effects "are clearly not essential to the phenomenon."
[2] The top five influences on wobble have been found to be
lateral stiffness of the front tire, steering damper, height of bike center of mass, distance of bike center of mass from rear wheel, and
cornering stiffness of the front tire.
[3][5]
An academic paper that investigated wobble through physical experimentation and computer modeling concludes: "the influence on wobble mode of
front tyre characteristics, front frame inertia and chassis stiffness were shown. In particular, it shows that [by] increasing front tire inflation, stiffness chassis, and front frame inertia about steering axis and decreasing sideslip stiffness of front tire, wobble mode damping is improved, promoting vehicle stability."
[6]
As this is not a common problem on FJRs I think you can safely discount all the stuff on chassis design / stiffness etc. Which leaves you with front tyre characteristics & increasing steering stiffness (hold on to the bars with at least one hand). YMMV