Convergence of the log-Sobolev constant in the central limit theorem

Let $\mu\in\mathcal{P}(\mathbb{R}^d)$ be an isotropic distribution, meaning that

$$\int x\,d\mu(x)=0\quad\text{and}\quad\int xx^T\,d\mu(x)=I_d.$$

Let $\nu_n$ denote the normalized $n$-fold convolution sequence from the central limit theorem, so that $\nu_n$ converges in law to the standard Gaussian distribution, and suppose that $C_{LS}(\mu)<\infty$. The log-Sobolev convergence conjecture.

$$\lim_{n\longrightarrow\infty}C_{LS}(\nu_n)=1.$$

Since $C_{LS}(\gamma)=1$ for the standard Gaussian $\gamma$, this would show that the log-Sobolev constant detects convergence to Gaussianity along the central limit theorem. The analogous convergence for the Poincaré constant is known, while the stated log-Sobolev convergence is proposed here and its status is not established in the supplied text.