Regarding the number of battles required, I'm gong to assert that the theoretical minimum is a pointless metric because so few people could ever achieve it. So let's consider a practical minimum from a player with an advanced roster that tries to be efficient. I'll use myself and a solid guildmate (https://swgoh.gg/p/685325334/) as benchmarks.
For C1-6, we both got max crate without any refreshes.
For C7, I used 11 refreshes. However, I finished a couple days early and wasn't as efficient as I could have been because Conquest was newly revamped. I believe my guildmate used 10 refreshes in C7.
For C8, he used 8 and I used 9. I finished 1.5 days early, so I could have been 8 had I been more patient.
For C9, we both wanted to do it in 7. IRL got busy for me, so I made some mistakes, and it wound up costing me 9 (only needed 1 battle from the 9th refresh). I don't recall his final count, but I do believe he fell short of 7, so let's say 8 or 9.
So I think it's fair to say that a practical goal for C7-9, for an endgame player that wants to be efficient, was 8-10 refreshes.
C10 is not done yet, so I can't give a definitive answer. There appear to be 22 combat nodes per sector, 110 total. We get 91 battles for free (6 per day for 14 days plus 7 for initial 144 energy). That's 19 extra battles.
All the MB nodes require 2 attempts to get both feats. At 2-3 keycards for 1 battle, redoing the MB nodes is very efficient. So that's 5 more battles.
Ditto for the S2 and S3 boss feats, so that's 2 more battles. S1 boss feats were both easy to get in 1 fight, so not counting them here. I'm considering the S4 GamGrd feat to be unviable for most, so I'm skipping that. And the S5 BAM+BK feat is either doable at the same time as the no GL feat or may very well be unviable.
I'm at 5 refreshes already, and I haven't talked about any of the feats. If we can only afford to pass on 32(?) banners, then we can't skip many feats. I already skipped 4 for the GG feat, so I have to do all but 1 of the Global Feats.
40 with BH and 20 with Boba+Fennec can be done simultaneously. Let's say you can get the 100 kills with Boba+Jango in the same 40 battles as well. That's 40 battles for 3 global feats. How many of these can be done while powering through the initial 110 battles? Based on what I've seen so far (early S3), I think calling this 20 extra battles is being VERY conservative.
I can only skip one Global feat. The 500 kills one is free. That means either Smugglers or Boba+Han. I think it's reasonable to say that the latter would be easier to earn passively than the former. If you're really on top of your game and doing the concurrent BH feats in the paragraph above, I think this would require at least 10 extra battles after completing S5.
I'm already at 10 mandatory refreshes. I haven't even looked at sector feats. I've assumed no losses or retries to get stars. And I'm assuming the ability to work on feats first pass at a rate much higher than I've been managing.
So practically speaking, until someone can finish C10 in fewer than 10 refreshes, C10 is grindiest Conquest to date. If someone does manage to do so, I will gladly recant and change my stance to it being grindier for me.
Hopefully this analysis is worth more than something Kyno saw on Reddit or Discord.