Well, compression is a major weakness when put before encryption.
There's inherent information leak in the form of compression ratio. Timing attacks work too.
I wonder if there's some work to mitigate them.

Bulat's branchless entropy codec looks like a full solution to the second for of leakage, though alone it will produce rather bad compression ratio. Has anyone considered branchless modelling?

Is there any research known to you on reducing the first form of leakage? Randomised compression?