fact that $S_n$ is the sum of independent random vectors is crucial only to obtain a good quantitative estimate for the quantities $|E[f(S_n)]-E[f(S_n^\dagger)]|$ for sufficiently smooth functions $f$. In the original CCK theory [@CCK2013; @CCK2017], such an estimate has been established by the so-called *Stein’s method*, especially *Slepian’s interpolation* (also known as the *smart path method*) and *Stein’s leave-one-out method*. Although their approach is not directly applicable to our problem, it suggests that we might alternatively use *Malliavin’s integration by parts formula* because it can be viewed as an infinite-dimensional version of Stein’s identity (cf. @SY2004). In fact, the recent active research in probabilistic literature shows a beautiful harmony between Malliavin calculus and Stein’s method, which is nowadays called the *Malliavin-Stein method*; we refer to the monograph [@NP2012] for an introduction of this subject. Indeed, this idea has already been applied in [@Koike2017stein] to a situation where $S_n$ is a vector of smooth Wiener functionals (especially multiple Wiener-Itô integrals) and $S_n^\dagger$ is Gaussian, which has produced several impressive results. Our plan here is to apply this idea to a situation where $S_n$ is a vector of multiple Skorohod integrals and $S_n^\dagger$ is conditionally Gaussian. In this regard, a relevant result 