Cache intermediary context values

The final resolution to #1100198: Context caching causes clearing problems was, essentially, "don't cache anything except the actual requested item". That solved the problem of caching data at the wrong level and therefore returning buggy data.

However, the net result is also that we now are far less aggressive in our caching, so intermediary values that could be cached safely are not. Oopsies. :-) This is a performance question.

We should try and figure out a way to cache those intermediary values.