Let $X_1,\dots,X_n$ be iid random variables with mean $0$ and variance $\sigma^2$, and let $$\xi:=\frac{1}{\sqrt{n}}\sum_{i=1}^n (X^2_i-\sigma^2).$$
Then, by the standard CLT, we have $\xi\Rightarrow N(0,v) $ as $n\to \infty$, where $v=\sigma^4(\kappa-1)$ and $\kappa$ is the kurtosis. Suppose I estimate $v$ with the consistent estimator
$$\hat{v}:=\frac{1}{n}\sum_{i=1}^n X^4_i-(\frac{1}{n}\sum_{i=1}^n X^2_i)^2$$
and do a right-tailed test at the level $\alpha$ for the following null and alternative hypotheses: $$H_0:E[X^2_i]=\sigma^2$$ $$H_1:E[X^2_i]>\sigma^2,$$ which means I reject $H_0$ if $\xi$ is greater than the $1-\alpha$ quantile of the $N(0,\hat{v})$ distribution.
Simulations with $n=1000$ reveal that the test is systematically undersized, which I think is due to the asymmetry of the function $x^2$. On the other hand the bilateral test against the alternative $H_1:E[X^2_i]\neq \sigma^2$ appears correctly sized on average. So one possibility would be to do a bilateral test in any case, but this sacrifices a bit of power against unilateral alternatives.
Are there other solutions? Thanks a lot for your help.
