It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way he proved that operads cannot be identified with monads. Now I'm wondering: is it true that every cartesian monad arises from an operad, i.e. every such a monad is the monad associated to an operad?

I little specification, what I mean here by operad is a generalized operad i.e. a $T$-operad for some cartesian monad $(T,\mu,\eta)$ in a cartesian category $\mathcal C$.