When the Specialise pass generates a specialised copy of a function,
it should transfer the INLINE information to the specialised function.
Otherwise, whether or not the INLNE happens can depend on whether
specialisation happens, which seems wrong.  See Note [Inline specialisation]
in Specialise.lhs

Here's the example Roman reported

    primWriteMU :: UAE e => MUArr e s -> Int -> e -> ST s ()
    {-# INLINE primWriteMU #-}
    primWriteMU = writeMBU . unMUAPrim
    ------
    
    The problem is that primWriteMU doesn't get inlined *sometimes*, which
    results in code like
    
    case Data.Array.Parallel.Unlifted.Flat.UArr.$sprimWriteMU
	    @ s11_X1nJ
	    marr_s25s
	    (GHC.Base.I# sc_s27F)
	    GHC.Base.False
	    new_s_a1Db
	    of wild3_a1Dd { (# new_s1_X1F9, r_a1Dc #) -> ...

Note the fact that we have a call to the *specialised* $sprimWriteMU.