It is well-known that a [commutative strong monad is the same as a monoidal monad][1].

Is there a notion of distributive law for commutative strong monads which is equivalently a distributive law for monoidal monads?

Does this follow from formal 2-categorical arguments of monads in a bicategory?

  [1]: https://ncatlab.org/nlab/show/monoidal+monad