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I have CoD Modern Warfare Warzone on PC and played yesterday and it hasn't updated on my exophase game card yet, is there no updating for this game on exopphase (under Blizzard/Battle.net)?
There is not, since the data isn’t public currently. Officially Blizzard only gives access to SC2 and WoW data for third party sites.
We found a workaround for Overwatch but for the other games like CoD the achievements aren’t shown anywhere online so it is impossible to pick them up as of now.