C'mon dude, you haven't even mentioned what kind of bike you are talking about or what tyres you are running. Finally 'Dunlops'. What? Dunlop Knobblies? Slicks? Are they in std sizes?
But yes as said above it was very common in the 80s to have a hands off wobble. Some bikes were very prone to it but it was exacerbated with mismatched tyres, weight over the back (loaded topbox or packrack) & mostly when the rear tyre got worn. (+ front tyre, steering bearings, alignment, suspension issues etc). Any of these can still be a cause on modern bikes.
Don't you look at my accountant.
He's the only one I've got.
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