To unlock those slots, you complete achievements/advancements or use the aforementioned trophies.
And don't worry, it'll work with other mods.
GitHub Page (bug fixes and translations welcomed): https://github.com/Lomeli12/TrophySlots
Stance on modpacks and videos: https://lomeli12.net/minecraft-mods/mod-packs
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Or see some of my other stuff on my website: https://lomeli12.net/