This following paper made the following statement without giving proof. https://www.jstage.jst.go.jp/article/jjss/35/2/35_2_251/_pdf/-char/ja
$$a^{20}=\frac{n}{p(n+2)}\sum_{i=1}^{p}s_{ii}^2$$ $$\alpha^{20}=\frac{1}{p}\sum_{i=1}^{p}\sigma_{ii}^2$$
$s_{ii}$ is the sample variance and $\sigma_{ii}$ is the population variance.
Under normality of data that generated the sample covariance $S$, $a^{20}$ is a consistent estimator for $\alpha^{20}$.
I'm not sure how to derive the distribution and/or prove the consistency of $a^{20}$.