
Valerion
VisionMaster Pro 2
Great picture, good gaming, but pronounced RBE and no lens shift.

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I have a Sony 715 and highly recommend it. The projector is quite large but it’s more serviceable as a result. The optics are best-in-class and it’s a true 3 chip system, so it doesn’t have rainbow artifacts.
I work in film post and highly recommend Sony’s SXRD projectors. They are native 4k, with 3-chips - so no color wheel or rainbows. Also, Sony’s lens optics are the sharpest I have seen. Sony’s can be calibrated to perfection but have very good color accuracy out of the box. The lamp-based VPL-VW665 and VPL-VW715 can be found used for your price range. The laser-based XW5000 is also a good option, they are newer but some are showing up open box etc.
Open box JVC or Sony will be your best option at that price. They are both true native 4K with 3-chip light engines - so no rainbow effect or pixel shifting. Personally I prefer Sony for their optics and image processing. I actually kept my first Sony 4K in storage after upgrading. It was a demo unit which I got for under $3K.
I went the open box route, got a 4K Sony and never looked back. Sony optics and picture processing are the best available and their projectors are true 3-chip systems and native 4k.
while im not contributing meaningfully to your question, sorry - i'd love to know, too. we have a LG C6 my uncle adores and a sony 4K 3D projector from 2020 that's quite frankly underwhleming compared to the LG C6. if there's any better while offering 4K/HDR, i'd LOVE to know to recommend to him
End of reviews

Valerion
VisionMaster Pro 2
Great picture, good gaming, but pronounced RBE and no lens shift.

Epson
Home Cinema 5050UB (V11H930020)
Best value, flexible lens shift, but dated tech, poor HDR.
Epson
Pro Cinema LS12000 4K PRO-UHD Laser Projector
Great 4K/120Hz gaming, no RBE, but needs calibration.

JVC
DLA-RS1200 / DLA-NZ500
Unmatched black levels for movies, poor for gaming.

BenQ
HT2060
Great 1080p gaming, deep blacks, but low brightness.

Ranked #1
Valerion - VisionMaster Pro 2
Ranked #1
Epson - Pro Cinema LS12000 4K PRO-UHD Laser Projector

Ranked #1
Hisense - Laser TV PX3-PRO

Ranked #1
Anker - NEBULA X1 4K Triple Laser Projector (D2351)

Ranked #1
Anker - NEBULA X1 4K Triple Laser Projector (D2351)

Ranked #1
JMGO - PicoFlix