Testing hypothesis about variance of non-normal population

Question

Let $X_1,X_2,\cdots$ be i.i.d. from a distribution $F$ with mean $0$ and unknown variance $\sigma^2$ and having four moments. A common test for testing $H_0:\sigma^2=1$ vs $H_1:\sigma^2>1$ is to reject $H_0$ for large values of $nS_n^2=\sum_{i=1}^n(X_i-\bar{X}_n)^2$, i.e., when $nS_n^2>\chi_{\alpha,n-1}^2$ where $P(\chi_{n-1}^2>\chi_{\alpha,n-1}^2)=\alpha$. Is his test asymptotically level robust if $F$ is non-normal in such a way that kurtosis $K=\frac{\mu_4}{\mu_2^2}-3$ of $F$ is non-zero?