Given an isotropic probability measure $μ$ on ${\mathbb R}^d$ with ${\rm d}μ\left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - α}}{\rm d}x$, where $α> d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$ is a continuous function and uniformly convex (${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$). By using Stein kernels for $\left( {α- d} \right)$-moment measures, we prove that the rates of convergence in the central limit theorem with sequence of i.i.d. random variables ${X_1},{X_2},...,{X_n}$ of the law $μ$, to be of form $c_{{\varepsilon}_0}\,\sqrt {\dfrac{d}{n}} $. The general case (i.e., $\varrho$ is only convex and continuous) remains open.