Injectivity of inherent scalar variance for projected normal variables on Stiefel manifolds

Let $X\in \mathbb{R}^{n\times k}$ be a normal random variable with $\mathbb{E}[X]\in \operatorname{St}_{n,k}$ and covariance

$$\operatorname{Cov}(X)=\sigma^2 \operatorname{id}_{\mathbb{R}^{n\times k}}.$$

The projected normal random variable is obtained by projecting $X$ onto $\operatorname{St}_{n,k}$.

Injectivity conjecture. The projected normal random variable has inherent average $\mathbb{E}[X]$. Moreover, there exists an injective function

$$\eta:[0,\infty)\rightarrow[0,\infty)$$

such that the inherent scalar variance is uniquely determined by $\eta(\sigma^2)$ for each $\sigma>0$.

This conjecture extends the corresponding known result for $\operatorname{St}_{3,1}=\mathbb{S}^2$ to general Stiefel manifolds. It is needed for parameter inference from projected observations, because an explicit formula for the projected normal density, and hence for the variance map, is difficult to obtain when $k\geq 2$.