Weak Convergence of centered and scaled sum to a non-degenerate limit implies existence of the second moments of the sequence

I'm looking at the following proposition:

$\text{Let}\, X, X_1,\ldots,X_n\colon (\Omega, \mathcal{A}, \mathbb{P}) \to (\mathcal{X}, \mathcal{B}) \,\text{be independently, identically distributed random variables, where $X$ is non-degenerate and}\, f\colon \mathcal{X} \to \mathbb{R} \, \text{measuarable. Then it holds that}$ $$ f(X_1) \in L^2(\mathbb{P}) \quad \text{iff} \quad \frac{\sum_{i=1}^n f(X_i) - n \mathbb{E}[f(X_1)]}{\sqrt{n}} \xrightarrow[n \to \infty]{d} X.$$

The implication from the left to the right is just an application of the classical central limit theorem, but the converse is interesting: Using theory about $\alpha-$stable processes we obtain, that $X$ must be normally distributed. Thus $X$ has finite variance, but I struggle to conclude, that the $f(X_i)$ have existing second moments.

Does anyone know the trick or any counterexamples? Kind regards.

Regarding the implication from right to left, since the normalize constant in the expression $\frac{\sum_{i=1}^{n}f(X_i)-n\mathsf{E}[f(X_1)]}{\sqrt{n}}$ is $\sqrt{n}$, so the law of $f(X_1)$ belongs to the domain of normal attraction of Gaussian distribution, then $f(X_1)\in L^2(\mathsf{P})$ is necessary. About the domain of normal attraction, please refer to B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of Independent Random Variables, Addison-Wesley Publishing Company, (1968), p181, Th 35.4 or I. A. Ibragimov and Yu. V. Linnik, Independent Stationary sequences of Random Variables, (1971), p92, Th 2.6.6.