ct.category theory - Are monads monadic?

In the book "Toposes, Triples and Theories", Barr and Wells study the question when a particular endofunctor admits a free monad. This is the case if the underlying category is complete and cocomplete and if the endofunctor preserves filtered colimits (we say that such a functor is finitary).

The question remains whether the resulting adjunction between monads on $C$ and endofunctors on $C$ is monadic. If $C$ is locally finitely presentable (lfp), this is true: Steve Lack showed in

"On the monadicity of finitary monads", Journal of Pure and Applied Algebra, Volume 140, Number 1, 21 July 1999 , pp. 65-73(9) (available here)

that the forgetful functor $mathrm{Mnd}_f(C) rightarrow mathrm{End}_f(C)$ from finitary monads on a lfp category $C$ to finitary endofunctors is monadic. Note that both these categories are again lfp categories, and we don't need universes to make sense of them, essentially because a finitary endofunctor is determined by what it does on a set of objects.

The result remains true if we consider categories and functors enriched in a complete and cocomplete symmetric monoidal closed category $V$ which is lfp as a closed category.

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