Doesn't matter the color spaces don't match, the color appearance should (and for me does). You do need to view both previews at the same size and at least zoomed to 100%.
I've gone by the book, 100% in Develop window, correct profile matching, etc. and to no avail.
However, good news. I think I have discovered the culprit.
This time, instead of using my own monitor profile, I went and used the original factory supplied monitor profile for the Spectraview. I then suddenly had perfect match between the two programs. I actually took a screenshot of both windows (the picture was a digital scale - black-midgreys-white) so I could measure the difference. And using the factory profile, the measument was identical in both Photoshop and Lightroom, like it should be.
I then swapped to my own created monitor profile, and the difference was measurable in the black patch only, the mid-grey and white patches were pretty much identical, or very close.
So my conclusion, some setting in the Spectraview profiler or the profiles it creates, interfere with the correct matching in Lightroom/Photoshop in the shadow area mainly.
(The difference is about 2-4 units, so not much, but visible).
Does this make sense?
Here is the difference