Shimmy in a nose-wheel OR tailwheel for that matter is indicative of a mechanical issue. Weight may or may not exacerbate the symptoms, but it is not the source of the problem.
But, to perhaps better answer your question, I would tell them to relax some pressure (but not all)......and have an A&P check out the tailwheel or nose gear.
I'm actually dealing with that right now in the 170. If I do a 3 pointer and have the yoke all the way back after touchdown, I get a shimmy. Relaxing some (but not that much) back pressure makes the shimmy go away. At least for now. I don't expect to have that problem anymore after the main leaf spring is changed out today.