Does a high ilvl ring/trinket also has a higher proc rate?

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For example does a higher ilvl Convergence of Fates also has a higher amount of procs?

If so, would upgrading my Ayala's Stoneheart Ring also increase the procs I get (if that makes sense)?

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Short answer: Usually not. In the case of CoF it is just the main stat that scales. Proc rates are usually constant over all ilevels and I can't think of an exception right now. If anyone knows one I would like to know.