The Unity team has already made some tests for their Engine with WebGL and WebAssembly and it seems promising: http://blogs.unity3d.com/2015/06/18/webgl-webassembly-and-fe...
Edit: I don't see how future addition of GC to WebAssembly changes anything I wrote above.
I replied to "Hopefully we will not need compiler to have high-performance C# engines in browser", because that does not make sense.
Unity3d does too many things (except new versions of C# :P) and ties to itself. There should be a direct C# -> WebAssembly compiler in future.