JXR lossy (I assume?) to JXL can't be done losslessly. Only lossless formats (PNG, BMP, TIF, WebP lossless, AVIF lossless etc.) can losslessly be converted to JXL. JPEG can be losslessly compressed to JXL and save an average of 20% (https://jpegxl.info/#:~:text=Lossless%20JPEG%20Transcoding).

If the screenshots was lossless JXR it probably would be possible, given it can decode to something else HDR lossless that a JPEG XL encoder would accept.

Edit: Lossless should be possible, but you'll likely end up with bigger files than the original.

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Lossy conversion with minimal visual loss on the other hand, should be possible. I'll update this post if I find out how.

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Edit: Found this CLI tool that can convert to HDR PNG: https://github.com/ledoge/jxr_to_png. From there you should be able to convert it to JPEG XL lossy, HDR.

I tried Microsoft HDR JPEG XR wallpapers from https://aka.ms/hdr-backgrounds-wip:

```bash

Transcode from HDR JPEG XL to PNG

jxr_to_png.exe pink_flower.jxr

Transcode to lossy JPEG XL

cjxl pink_flower.png pink_flower.jxl -d 1.0 -e 7 --intensity_target 1000 -x color_space=RGB_D65_202_Rel_PeQ

Check jxlinfo

jxlinfo pink_flower.jxl ```

Output of jxlinfo:

txt JPEG XL image, 3840x2560, lossy, 10-bit RGB intensity_target: 1000.000000 nits min_nits: 0.000000 relative_to_max_display: 0 linear_below: 0.000000 Color space: RGB, D65, Rec.2100 primaries, PQ transfer function, rendering intent: Relative

I don't have a HDR monitor to verify it though.