Looks like we've pinned it then.My ilevel is 367 and I was rewarded with a 370.
:/ Reading through the previous comments I still have no idea how this works. My husband and I both play druid, both have 361 item level equipped. Both have Azerite level 28 neckpieces. Yet he gets the 370 azerite caches and I only get the 355 ones. Helm/Chest/Shoulders for both of us are 370/375/375. I can't figure out why he gets the higher one and I still get the lower option. The only difference I can find is that I don't have the reputations exalted and he does - but I don't see anywhere that says that's a factor.
My Pally is 362 ilvl and got a 370 piece from the cache. 370 Shoulder, 355 head, and 355 chest. My hunter is 362 ilvl with the same ilvl pieces and is still at 355...WTH!
Just want to share what I have experienced. I used to have 3x355 (with overall iLvl ~360) and azerite necklace at 32(=387 iLvl on it). At all times I got 355 but nothing higher. Then I looted a 385 chest, after which I started to get 370s from cache. My guess it is all related to only these 4 pieces and their iLvl, which in average should be >=367 (x4=1468 in total).
351 just got a 370 from emissary daily reward
Can you get rings, trinkets, off-hands or weapons from these?