Optoma launched the 4K DLP projector UHD55 which allows low-latency screen projection. It is a 3600 lumens projector with a resolution of up to 3840 x 2160 pixels whilst offering low latency and be... Optoma Releases low latency 4K Projector with 240 Hz Full HD game mode for $ 1799
https://hometheaterreview.com/optoma-uhd55-new-3600-lumen-gaming-and-movie-projector/ I would be worried about power consumption @3600 lumens if also using as a large TV, with the huge bump in recent electricity prices and worse coming. It can only provide 95% DCI/P3 coverage at reduced brightness levels. Its not really 4K at 60Hz due to the pixel shift. Though lamp life isnt bad at 4 to 10,000 hrs, it depends on the replacement cost. It does 3D, that might make it worthwhile, if only gaming support hadnt quit the technology.
TronicsFix recently repaired a bunch of Optoma projectors. The flaws were quite strange, like loose internal connectors and screws. I don't know how good and respected, generally speaking, Optoma is, but that was a pretty weird repair video.