A WoW player shared a post on Reddit showing that the game looks at what’s in your Mythic+ dungeon chest before you even open it.
Reddit user electrikmayham was doing Cinderbrew Meadery, and the only non-Hero/Mythic piece they had left was the neck. As soon as the group beat Goldie, the final boss of the dungeon, the player received a lot of achievements, even though they hadn’t touched the chest yet. Obviously, the neck piece was found inside the chest.

This serves as proof that the game first checks M+ chests for achievements even before you touch it.