As he says, it is the stock that is moving, not the barrel in the action.
I have wrapped HD tape around the shank of a stock bolt where it rides in the wood of the wrist; a real buttstock bolt, rather than common bolt, would have threads no larger than the shank dia. and the hole through the wood should be just over shank dia. The H&R hole is oversize, so all you are getting is a pinching action between the washer under the head and the treaded action, no support from the hole through wrist, thus play can be noticed. The tape wrap did tighten this up for me, and even though it was a quick fix for trials it is still there.
I have found that vertical stringing is often caused by inconsistent placement of the buttstock to the shoulder, especially with heavier kickers. This will be accentuated with a loose bolt hole to bolt fit. Competition rifles would never be made with that much tolerance and depend only on pinch pressure to hold in place, unless the fitting to the action had a mortice to lock the wood in place.