I have a Virgin V6 box, a high quality 4K HDMI lead, a Samsung TV and an Optoma Projector. The TV and projector are both 4K and with HDMI HDCP V2.2 inputs.
When I connect the HDMI lead from the V6 to the Samsung TV, I successfully get a 4K / Ultra HD picture.
When I connect the same HDMI lead from the V6 to the Optoma Projector, I only get an HD picture (1080). When I plug other 4K / Ultra HD sources (e.g. Amazon Firestick, Disk player) into the projector using the same HDMI lead (without involving V6 at all), I successfully get a 4K / Ultra HD picture.
I have tried the 'Test Formats' option on the Tivo V6, it does not think the 4K projector is a 4K projector!
I'd suggest the issue is HDCP 2.2 relatated. The V6 has a very basic implementation of HDCP that can and often does throw a wobbly. The STB is obviously not correctly handshaking with the PJ and isn't detecting the PJ's HDCP 2.2 compliance. You'd only be able to convey commercial 4K encoded video to a device if the V6 detects that it is HDCP 2.2 compliant. You are unfortunately at the mercy of Optoma and or Virgin Media to rectify this and it would require a firmware revision on either the STB, the PJ or both. I'd not hold your breath in expectation of Virgin Media issueing such a firmware revision.
Out of interest, how long is the HDMI cable from the V6 to the PJ? If longer than 5m then I'd maybe also suggest you consider using HDMI over fiber as opposed to conventional copper HDMI cabing. You also say that the V6 is connected to both the PJ and ta TV. How are you doing this and is there an intenmediate splitter or other device between the source and your PJ?
A different HDMI cable would be my first bet. It may be it is not the 4K bit you are having issues with but the HDCP signals. A friend of mine has an Optima and could only get the Optima do full 4K/HDR on everything by getting a fibre HDMI cable - even though they already had a high quality cable.
Did you select 1080p on the V6 or did the box automatically set it? What happens if you manually select 2160p?
What happens when you switch HDR and the HDR SIM setting on/off in the projector settings and try for 2160p mode on the V6?