Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on $\mathcal{C}.$

Assume that the model structure on $\mathcal{C}$ lifts to a model structure on the category of $\mathrm{T}$-algebras $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}),$ where the weak equivalences and fibrations are those of $\mathcal{C}.$

Assume that $\mathrm{T}: \mathcal{C} \to \mathcal{C} $ preserves homotopy colimits indexed by $\Delta^{\mathrm{op}}.$

Does the forgetful functor $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}) \to \mathcal{C}$ preserve homotopy colimits indexed by $\Delta^{\mathrm{op}}?$

More generally one can ask the question replacing $\Delta^{\mathrm{op}}$ by an arbitrary category.

Remark: If $\mathrm{T}: \mathcal{C} \to \mathcal{C} $ preserves colimits indexed by some category $\mathrm{K}, $ the forgetful functor $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}) \to \mathcal{C}$ preserves colimits indexed by $\mathrm{K}.$

This also holds for $\infty$-categories with the appropriate notion of monad and algebras over a monad.