I thought it was just the crystal tower raid for 5.3
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The MSQ requires the Crystal Tower series. The ShB relics/Bozja quest lines require the level 70 Return to Ivalice raids.
HW is standalone and doesn't connect to anything else (yet) but they're still worth doing because sky pirates
Crystal tower is extremely storyline-relevant for Shadowbringers. It's also required to continue the MSQ.
The Mhach raids explain a bit about the Fifth Astral Era, but are kind of a side-story. Nothing else requires finishing them.
Ivalice is relevant to Bozja, and required to start it. Both say a bit about what Garlemald is doing in other provinces. There was an actual press release saying Ivalice's story is over and done with though. Despite Bozja ending on a cliffhanger.
The NieR raids have nothing to do with anything else - just a weird crossover.