There has been ongoing work toward supporting this feature. See:
http://juliacomputing.com/blog/2016/03/10/j2c-announcement.h...
How close are we to web assembly compilation?
I'm not aware of anyone working on it. There were some attempts with Emscripten which ran in to unsupported instructions; some of those may be supported by newer versions of Emscripten, but I haven't heard of any attempts to get this going recently.
Gotcha. Is it definitely technically feasible, or is that still unknown?
For Julia core it should be feasible, but possibly requiring patches to Emscripten, so the level of effort is still unknown. The other issue is the C and Fortran numerical libraries -- some of which use inline-assembly, others (Fortran) don't have a working LLVM frontend as far as I know.
Rumours persist that this feature is planned. Or are they just plain wrong?