Deriving Functor / Foldable / Traversable for co-pattern type

This is probably asking too much, but could GHC derive Functor, Foldable ... for this co-pattern data type?

data StreamTag :: Type -> Type -> Type where
  HeadTag :: StreamTag a a
  TailTag :: StreamTag a (Stream a)

newtype Stream a = S (forall res. StreamTag a res -> res)

instance Functor Stream where
  fmap :: (a -> a') -> (Stream a -> Stream a')
  fmap f (S stream) = S $ \case
    HeadTag ->      f (stream HeadTag)
    TailTag -> fmap f (stream TailTag)

instance Foldable Stream where
  foldMap :: Monoid m => (a -> m) -> (Stream a -> m)
  foldMap f (S stream) = f (stream HeadTag) <> foldMap f (stream TailTag)

I'm guessing it's not worth it in any case, but is it uniquely determined?