What are the best loudspeakers under $4000 to re-create lifelike piano

This is easy.  Build the Linkwitz LX521 system.  Easy to do for $4000 or less.  And that includes amplification.

Siegfried Linkwitz is one of the premier audio techs on the planet, so you already know his design is superbly engineered.   It doesn't take a genius or $1 million worth of shop gear to build them.  

Build the LX521 and have something equal to the big $60k systems you see here.

@snapsc- I would venture that if a system produces piano convincingly it ought to do most everything else well, given not only the range of the instrument, but the other demands placed on its reproduction-- dynamics, power, timing, harmonics, etc. 
When I listen for evaluation purposes (as opposed to enjoyment), I try to use a range of different material, almost none of which is "audiophile"-- I don't mean that in a pejorative way, just that I don't want records that are known sonic spectaculars to find the weak points- i want to hear what a system can do "on average" since much of what I listen to was never released in audiophile quality issues though some recordings are still pretty killer. (and some aren't). 

Pianos recorded in different venues by different engineers sound different. Period. I find that the wider the range of my system (using a couple of subs) the more enjoyable pianos sound, but all are different (did I already say that?). I recently worked as the sound mixer for an Eldar show (a physical young piano genius with a hot little trio), and after setting up the 2 condenser mics I generally use for piano…lid open…he thought we should instead try a large diaphragm single condenser I had hovering over the drums, and stick it in the piano with the lid closed as much as we could close it…sounded great. A unique approach to get what the artist wanted and some EQ to keep the piano mic from low end feedback and there we went. Was it authentic? I heard Brad Mehldau at a hall near Harvard recently playing un-miked, and although the hall has decent acoustics as far as that goes, much of what he was doing was inaudible to much of the venue…authentic, musically brilliant, too quiet for that room. I had a great time anyway.

real_music77712-15-2017 New WaveTouch Audio Antero ! 😍😇
https://www.youtube.com/watch?v=NuEIeN46ZMc 

Thanks for the Link!
Another beautiful piano performance.  Once Upon A Time In The West (Steffi Vertriest)   Pianist is moody.