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I recently had a similar problem prepping my bike for a track race. I used 4 shims from a soft drink can. I cut each about 1 inch by 3/4 inch with a small bit of each corner snipped off (this made it easier to slip into the stem opening). I started with one and kept adding until I got a firm, solid grip. No need to wrap the whole handle bar or use tape. The bars held solid through several induced applications of adrenalin.
Are these still available as I am very interested? Thanks