Simplifier: Insufficient eta expansion of arguments

While hunting regressions in !9874 (closed) I found that T21839r regressed by 10%. I realised that the argument in the following program is not properly eta-expanded:

g :: ((Integer -> Integer) -> Integer) -> (Integer -> Integer) -> Integer
g f h = f (h `seq` (h $))

After simplification, we have

g = \ (f_aiR :: (Integer -> Integer) -> Integer)
      (h_aiS :: Integer -> Integer) ->
      f_aiR
        (case h_aiS of h1_X0 { __DEFAULT ->
         $ @GHC.Types.LiftedRep @Integer @Integer h1_X0
         })

It would be much better to eta-expand the Case expression and subsequently inline $.

I believe is this due to the Simplifier never checking whether it can eta-expand an argument.