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

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

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

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

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?