Let $S$ be a monad on a category $\mathbf C$. If $\mathbf C$ is cocomplete, and the category of algebras $\mathbf C^S$ admits reflexive coequalisers, then $\mathbf C^S$ is cocomplete. Thus, it is existence of reflexive coequalisers that is the central question in establishing cocompleteness of categories of algebras. There are various theorems giving sufficient conditions for categories of algebras to admit reflexive coequalisers (for instance, see this nLab page). For the purposes of this question, let us assume that the categories involved are locally presentable, and the monads are accessible (i.e. preserve $\kappa$-filtered colimits for some cardinal $\kappa$).
Now suppose we have a lax morphism of monads $(F, \varphi) : (\mathbf C, S) \to (\mathbf D, T)$, comprising a functor $F \colon \mathbf C \to \mathbf D$ and a natural transformation $\varphi : TF \Rightarrow SF$ satisfying compatibility conditions with the units and multiplications of $S$ and $T$. Recall that such morphisms induce functors $\mathbf C^S \to \mathbf D^T$ between the categories of algebras that commute with the forgetful functors.
Question. Are there reasonable sufficient conditions on a lax morphism of monads $(F, \varphi)$ for the induced functor $\mathbf C^S \to \mathbf D^T$ between the categories of algebras to preserve reflexive coequalisers?
"Reasonable" is up to interpretation: I would expect at least for $F$ to preserve the limits and colimits in $\mathbf C$ necessary to form reflexive coequalisers in $\mathbf C^S$ (potentially all limits and colimits). However, restricting to strict/pseudo morphisms (i.e. asking for $\varphi$ to be invertible) would be overly restrictive.
