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Memory Dungeons (メモリーダンジョン) are a Permanent / Training Mission Type released alongside Flower Memories on September 2, 2019. Clearing these stages gives you a set amount of Memory Crystals (メモリークリスタル) which can be used to buy Wares.
In the past, there was another reward called Memory Stones. However, they were removed from the game on January 12, 2021.
Memory Dungeon[]
Memory Dungeons (メモリーダンジョン) are a generic type of Memory Dungeon that relates to some character's memory. These missions reward much more Memory Crystals than the Oath Memory Dungeons and should be the main source of farming them.
Oath Memory Dungeon[]
Oath Memory Dungeons (誓いのメモリーダンジョン) are a special type of Memory Dungeon that reward Eternal Oath items. It was released alongside Eternal Oath on January 25, 2021. Clearing these stages earns you a set amount of Memory Crystals plus chest drops that may include Bond Oaths (Silver, Gold, or Rainbow) and Petite Oath Hearts. x100 Petite Oath Hearts can be refined to create a regular Oath Heart. These stages offer less Memory Crystals than regular Memory Dungeons (12 at most), however they drop Eternal Oath items.