I’ve noticed that they Disney+ app does not have a 4K SDR capability. It either outputs 4K HDR/Dolby Vision or 1080p SDR. So if the display is SDR or you set your player to SDR (eg. Firestick Max or ATV4K), you will get a 1080p movie from Disney+
You can see this behavior by turning on developer tools on a Firestick Max. Below is a image from Disney+ where I have set the Firestick Max to 4K SDR and Disney+ is sending at 1080p file
Because low brightness display devices like our projectors don’t have enough dynamic range to handle a HDR signal well, which results in overly dark scenes or clipped highlights. Hence, if you can receive a 4K SDR stream, the images a brighter and more balanced.
Of course a Lumagen or MADVR will tone map the HDR range into a range appropriate for the projector, but it costs more than most projectors.
If I find a HDR movie too dark e.g., Captain Marvel, I switch to a SDR stream and it looks much better. The problem with Disney+ is that it doesn’t have 4K SDR streams and will switch to 1080p. The iTunes store has no issue. When I turn off HDR, I get a 4K SDR stream
Noted on the limitation of the Disney + app. But I thought the LG projector you are using right now is pretty good on the lumen output to display a decent HDR image.
The LGHU810 is for most content, but I have the same HDR darkness issue with my Panasonic OLED, which peaks out around 650 nits. The dynamic tone mapping on my OLED is decent, but there are still some Disney+ flicks like The Mandalorian, which look much better in SDR, even at 1080p. The Vertex2 helps, but The Mandalorian is too far down the dark HDR curve to be pleasant in HDR.
JVC projectors can have custom tone curves loaded for their frame adapt HDR. I will be experimenting with Calman’s custom tone curves on the LG projector at some point to see if I can brighten up dark HDR content.
If you want to see what I mean, I think you can switch between SDR and HDR on your Nvidia Shield. Compare the difference in Disney+ between HDR and SDR.
You are talking about dynamic tone mapping here in short. And yes, HDFury Vertex 2 can only do so much and this is the reason why I sold off the Vertex 2 and opted for the Lumagen Radiance Pro and never look back. JVC HDR FrameAdapt is the closest alternative to getting a good DTM (come built-in) but the on the fly scene-to-scene and frame-to-frame DTM is meant for their range of product. The most atrociously priced MadVR is another product which in capable hands can produce amazing HDR imagery. At some point, you have to decide what you want…if you are into 4K HDR content and needed a Video Processor that can act as a calibration toolbox as well as doing DTM by auto-converting HDR to SDR whilst retaining the BT2020 colorspace, I’ll say go for the Lumagen and don’t look back. That’s the easiest route to take…of course, if you have the time and energy, you can experiment with the Calman’s custom tone curves to see any tangible benefits when it comes to 4K HDR content.
The Lumagen at $6-10k is twice the price of a typical projector that I would buy So, unless you have a $20k plus SXRD projector it doesn’t make sense for most of us.
Two simple solutions to map some dark HDR movies into something a projector can display well.
Convert to SDR. Set the ATV4K to 4K SDR or set a Firestick Max to “disable HDR” and use the Apple TV app. If you play an iTunes Dolby Vision movie, Apple will convert the stream into a 4K 10bit SDR Rec 709 stream before sending it to your ATV4K or Firestick Max. While you don’t get colors beyond Rec 709, you get smooth 10 bit color (less banding) and 4K resolution with bright images.
Get a HDFury device like a Vertex 2 or the $300 Arcana. With that you can play a Dolby Vision stream tone mapped to your lower nit projector using technology by Dolby Labs and the Movie Director encoding more information into the Dolby Vision stream. If you set it up right, it actually looks quite good. The Dolby Vision stream has brightness information (metadata), which allows the player’s processor (like the ATV4K) to tone map accurately without high processing power. In contrast the Lumagen and MadVR require enormous processing power because they have to take a non-DV HDR10 stream, which has no metadata and analyze each frame in order to tone map it to the lower nit projector. That’s why they cost a lot of $.
Many streamed movies are not that colorful, so the SDR Rec 709 movie will look the same as the HDR BT2020 movie. Exceptions would be colorful movies like Aquaman. Enabling your projector to read Dolby Vision metadata through LLDV works quite well, but a few movies will still not look very good unless you tweak the HDFury settings.
For UHD Blu Ray, the Panasonic Blu Ray players eg. UB820 tone map well.
Thanks for sharing all those details. And by all accounts, you are spot on and I have nothing further to add on to what you already know. You did your homework well.
So, unless you have a $20k plus SXRD projector 
it doesn’t make sense for most of us.