Identity of the Fakir idempotent approximation for classical filter monads

Determine whether, for the following classical monads: (a) the ultrafilter monad on the category of topological spaces and continuous maps, (b) the prime open filter monad on the category of topological spaces and continuous maps, and (c) the open filter monad on the category of T0 topological spaces and continuous maps, the first inductive step in the Fakir idempotent approximation of a monad—namely, the idempotent monad I_T whose underlying functor T^ϕ is defined as the equalizer of Te and eT—is isomorphic to the identity monad.

Background

The paper studies monadic aspects of the ideal lattice functor and uses the Fakir construction to analyze when iterations of monads/comonads stabilize. A key theme is identifying when the first inductive step in the Fakir idempotent approximation of a monad becomes the identity monad, which yields an equivalence between the ambient category and the category of free algebras.

For distributive lattices with the ideal functor monad, the first Fakir step is essentially the identity monad, leading to the equivalence between distributive lattices and coherent frames. The authors then consider classical topological monads (ultrafilter, prime open filter, and open filter monads) that satisfy the conditions for their general equivalence theorem, and note that Jacobs’ result applies in the lax idempotent cases.

However, the authors explicitly state that, for these classical monads, it is unknown whether the first inductive step of the Fakir idempotent approximation reduces to the identity monad. Resolving this would parallel the distributive lattice case and potentially yield representation-type equivalences between the ambient topological categories and their free algebras.