Very simple and stripped-down Rust programs have already been demonstrated to work via Emscripten. I expect the work here would make it a little bit easier, but I don't know of anyone actively working on supporting this use case.

Last I heard, more advanced support hinged on Emscripten upgrading their version of LLVM, which was much more ancient than the bleeding-edge version that Rust uses.

I was wondering that myself. I remember it being given as the main blocker when someone asked about Emscripten, but that was over a year ago and a lot has changed since.