How do the Nemesis upgrades work for uniques?

In particular, once I see the unique get either LP or an affix, can it roll another LP or an additional affix? The only thing I’ve seen so far is upgrading the affix by some amount of tiers. If I open up the Nemesis window and see that the unique I empowered got some mod I don’t want or only 1 LP, do I just stop there and go roll for another one or is it possible for me to empower again to gamble for something extra?

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I’ve yet to see a unique get LP, but they can definitely get more than 1 mod with multiple empowerments (no idea if you’re in a high enough mono/corruption whether they could roll multiple affixes on the first empowerment).

LP is a rare outcome, I’d say somewhere between 10-20 %.

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I’ve had one get LP (a common one) and I’ve had 2 get an affix (so become legendary).

I’ve had a few get 1LP (including a Bastion of Honour), more have gotten added affixes.