Looking for an integrated amp around 4000 to 5000.

@Pal, what is the Rogue not doing in your system? Have you rolled 12AU7s to better serve your wants?

You can pick up a used Naim Supernait 2 for relatively cheap now that the SN3 has just been released, but even a new SN3 would fit in your budget.  Won't have balanced input, but it'll do you right in engaging you with the music, is built like a tank, and drive most speakers you are liable to throw at it (should be fine at 88db sensitivity with yours).

I own the PrimaLuna Evo 400 and it sounds beautiful with my Salk Sound Song3 speakers. I have heard the Mark Levinson and thought it was a good amp also.

Look into Octave. I heard a v110 pushing some dynaudio Contour 60s and it really dominated them. Clean clean sound with a little tubeyness and some balls. 

The Ayre AX5-Twenty was the most amazing integrated I’ve heard. I got a VX5-Twenty amp and pair it with my BAT preamp but the integrated sounds better at least in my system and with my musical tastes. I believe you can find a used AX5-Twenty in your price range.