Hi.
Are there plans to at least make it possible to access full-sized JPGs/PNGs manually? For example, this online tool allows you to embed secret messages/images into a seemingly random PNG image.
Here’s an example image that outputs Hello World. It’s 4x4 pixels in size, so you may need to zoom in to save it.

This image won’t get compressed by Twitter’s servers, at least I hope, so it should be ok.
And here is another example image that outputs a much larger image with the text “Hello World”:
The second image would be rendered unreadable if converted to JPG, as the data embedded into the picture is literally within the pixels. As you mentioned earlier, many people don’t have the bandwidth to download hi-res images all the time. However, perhaps it would be possible to still host the image at its original size, just not to users on their Twitter’s feeds? (In fact, I do believe it’s something you already do, just not for PNG images. https://pbs.twimg.com/media/DrbHwt7U8AA6X05.jpg:orig)
Edit: in case this needs clarification, here’s what Bandcamp does for album covers…
Publicly facing image (shown to the user on an album page): https://f4.bcbits.com/img/a2631671759_10.jpg
Full-sized image (not shown to the user, but still exists on the server. even if the actual image is not a png, it’s still a good idea to add it so you ensure you get a high quality image): https://f4.bcbits.com/img/a2631671759_0.png