My understanding is that both pace at your starting AeT HR should go up and your heartrate at AeT should go up (and be even faster still). As an example, for me, I started out at 145 bpm and my pace was around 12 minute miles. After a few months of consistent training, both things improved, 1) my pace at 145 bpm dropped to around 10:00 minute miles; and 2) I re-tested at 10 bpm faster at 155 and I was just under 9 minute miles.
It’s been a few months since that second test, and I think my AeT heartrate is has maybe gone up a tad in that efforts at 155 feel slightly easier than they did, but I’m certainly not cruising at 10 bpm faster and nose-breathing still. But I am fairly confident if I did another test at 155 my pace would be even faster than ~9 minute miles. In other words, my pace at a given heartrate has continued to get better, but the AeT heartrate “improvement” is more marginal. Or, at least dropping 30 seconds off my mile time at a given heartrate seems more material than maybe adding 2-3 bpm to my “AeT heartrate” given that some days feel easier than others. Hopefully that makes sense. I’m not sure how common my experience is though – I’m sure the coaches will have a better answer of how to measure improvement.