Sucks pressing on the emoji picker and then the whole website times out because you spammed it. Also sucks waiting for it all to load.
There is a cache-control header but it doesn't seem to do anything... maybe there's missing etag or modified-since on those images? Aslan it should just be perma-cached.
Looking at the headers and checking the docs, everything seems like it "should" work. I would bet it's some JS that isn't respecting the caching.