Log-normal approximation for the Wasserstein normality-test statistic

Let $Y_1,\ldots,Y_n$ be independent identically distributed samples from a Gaussian distribution, let $\mathbf{y}=(Y_1,\ldots,Y_n)$, let $\bar{\mathbf{y}}$ be their sample mean, and define the reduced sample vector $\tilde{\mathbf{y}}$ as

$$\tilde{\mathbf{y}}=\frac{\mathbf{y}-\bar{\mathbf{y}}}{\sum_{i=1}^n(y_i-\bar{\mathbf{y}})^2}.$$

Let $\mathbf{a}$ be the reference vector used in the Shapiro–Wilk statistic, and let $\mathrm{W}_2^2(\mathbf{a},\tilde{\mathbf{y}})$ denote their squared Wasserstein distance. Log-normal approximation conjecture. If the $Y_i$ are normally distributed and $n$ is sufficiently large, the distribution of $\mathrm{W}_2^2(\mathbf{a},\tilde{\mathbf{y}})$ can be approximated by a log-normal distribution $\mathcal{LN}(\mu_n,\tau_n)$. The conjecture is intended to provide a decision threshold for a Wasserstein-based test of normality; the source gives no evidence resolving the approximation.