Is a pseudonatural transformation of strict 2-functors to Cat isomorphic to a 2-natural transformation?

Let $\mathcal{C}$ be a strict 2-category. A corollary of the bicategorical Yoneda lemma says that any pseudofunctor $\mathcal{C} \to \operatorname{Cat}$ is pseudonaturally equivalent to a strict 2-functor. I would like to know if the "next level" of strictification is true; namely, is it true that any pseudonatural transformation of strict 2-functors $\mathcal{C} \to \operatorname{Cat}$ is isomorphic (via an invertible modification) to a 2-natural transformation?

The specific case of this that I am interested is when $\mathcal{C}$ is the delooping of a monoidal category: the first statement says that any left $\mathcal{C}$-module can be made into a "strict" left $\mathcal{C}$-module, while the second would imply that any morphism of left $\mathcal{C}$-modules can be strictified.

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It is, however, true that any strict (or even pseudo) 2-functor $F:C\to Cat$ is pseudonaturally equivalent to a strict 2-functor $F':C\to Cat$ which has the property that any pseudonatural transformation $F'\to G$ is isomorphic to a strict 2-natural transformation. An $F'$ with this property is called flexible (if you're a 2-category theorist) or cofibrant (if you're a homotopy theorist). The possibility of flexible replacement follows from generalities about 2-monads: there is a strict 2-monad T on the 2-category $Cat^{ob(C)}$ for which strict T-algebras are strict 2-functors $C\to Cat$, strict T-morphisms are strict 2-natural transformations, and pseudo T-morphisms are pseudonatural transformations. The general coherence theorems of Power and Lack (see the papers "A general coherence result" and "Codescent objects and coherence") apply to this 2-monad and specialize to the statement I quoted above. The homotopy theorist can instead construct a model structure on the 2-category of strict 2-functors and strict 2-natural transformations in which the cofibrant objects are cofibrant/flexible; see Lack's paper "Homotopy-theoretic aspects of 2-monads."