Before you install the new petcock try that same hill with the lever in PRIme.
It's possible that you have the throttle wide open, especially if you are in a higher gear, which would leave no vacuum to keep the petcock open.
If it works in PRIme, and you are trying to go up that hill in fourth or fifth gear, try it in ON and in third gear to see if it makes a difference.
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Yano Steve, something is wrong with your theory. I have pinned the throttle all the way up hundreds of long grades, several thousand feet vertical at a time, never had this happen. Usually it's in whatever gear puts it in the 7,500 - 9,500 RPM range, but sometimes less, maybe as low as 5,000. On the bigger bikes maybe even lower. I think even with the throttle wide open there's enough vacuum to keep the petcock open.