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

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I'd increase the size of that screen and maybe go with something like a benq 2720i which also does 3D. If you go the large tv route, can you easily get it in there? Up or down stairs ("Pivot!"), through doorways, have a large truck bed? Remember, one day you may have to get it out of the house. Do you have a good recycling center?
There are not many projectors that fit your requirements but here's what I would recommend. No short throw projectors (other than the valerion max) have lens shift, so we need to be precise when mounting. Are you going to be mounting? I can tell you exactly where to mount it if you are. I usually recommend the Benq W2720i, but it does a 135 inch image at 9' 8'', and the projector will make it outside of your 10 feet. \#1 - Hisense C3 Ultra \#2 - Hisense C2 Ultra \#3 - Benq TK710STI
While my projector doesn't support 4k 120 Hz natively, it does try its best. I have the BenQ W2720i, and it looks a lot better than 1080p 60/120/240, and a lot better than 1440p 60/120 in games. Outside of games, 4k 60 looks best and it's not even close, especially when just browsing. I'm looking forward to native 4k 120/240 Hz DLP. Hopefully we're not too far out.
I just bought mine! :) I can't use a UST unfortunately. It does 1080p 240, but I can see about 50 feet in front of me in games.
Why isnt benq w2720i in this list? Pretty happy with it myself, around $2000
I recently picked up a BenQ w2720i. The pseudo-4k image looks excellent in my opinion. The colours and focus are great, too. However, I cannot tolerate the rainbow effect! This is my first DLP projector (coming from a Nexigo PJ40) and it turns out I am very sensitive to RBE. I see it every time I blink and during any fast screen movements and subtitles. I'm within the window to return this projector and will do so. What I'm looking for are suggestions for projectors that have * zero rainbow effect; * led or laser light source (I don't want to be changing bulbs); * 4k-ish output. So far the only projectors I see that promise zero RBE are Epson projectors with a traditional lamp. The BenQ satisfied 2 out of 3 of those requirements. If I can't find anything I may fix the burnt screen on the ol' PJ40 and put it back into service.
I find ECO mode to produce an appropriate brightness. The screen is a generic pull-down. It must be vinyl or something. I picked it entirely because it could retract and be easily cleaned. I didn't expect that a screen could be part of the RBE issue. EDIT: And all that said, I think the picture looks excellent! The image quality is perfectly to my liking; minus the RBE...
I checked the firmware. The w2720i is already on the latest version (1.02) that is offered. This evening I messed around with practically ever display related setting I could find, including different refresh rates. I could only seem to make the RBE even worse with my tinkering, so I put things back to mostly default, except for the ECO mode brightness and "Film Maker Mode" picture mode.
I upgraded to my new BenQ W2720i and I love it! The picture quality is awesome, it’s silent, and the best you can get for this money. I also got a discount from CB. Just make sure you have the right distance!

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