Let $\mathcal K$ be a bicategory. For every pseudomonad $T : \mathcal K \to \mathcal K$, does there exist a 2-monad $S : \mathcal C \to \mathcal C$, where $\mathcal C$ is a 2-category biequivalent to $\mathcal K$, such that the bicategory of pseudoalgebras $T\text{-}\mathrm{Alg}$ is biequivalent to the 2-category of pseudoalgebras $S\text{-}\mathrm{Alg}$?
That is, can we always choose to work with 2-monads rather than pseudomonads, so long as we consider their pseudoalgebras, rather than their strict algebras?
If this is true, does the statement continue to hold for the 2-category of strict algebras $S\text{-}\mathrm{Alg}_s$? (Not every pseudoalgebra is equivalent to a strict algebra, but this fact does not take into account changing the 2-monad: I imagine this is false, but it would be good to have a reference.)
