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But then they would need a new domain.

Not really. gfycat does both gifs and webm.

It would also be more efficient to be able to tell when the gif started

Changing the desktop background to a solid color would help too.

Doesn't GIF's difference encoding mostly make the background a constant cost?

It would make the initial frame smaller, but the difference on following frames should be negligible yes.

MS Edge supports WebM, and that's basically the current version of Internet Explorer.