The first three tasks (Deep Caverns Collector's Case, The Bone Rod, Lost Wayfarer's Tent) get a check mark after completing. But after logging out and in again, they disappear. Even the reward parts are in bank or on character. It is not possible to get the achievemnt done. Only task that keeps its check mark is Extruded Black Pearl.
I believe someone on discord posted about a similar (or identical, idk) issue, and they fixed it by placing the items in trade window with a pet and then cancelling the trade.
This might have been it: https://discord.com/channels/618902729925984279/621399777028079637/832448182927097916 https://discord.com/channels/618902729925984279/621399777028079637/833392102624329778