Small-variance conjecture for projected normals on Stiefel manifolds

Let $\mathbb{K}$ be the underlying scalar field, let $\operatorname{M}_{n,k}(\mathbb{K})$ be the ambient matrix space, and let $\operatorname{St}_{n,k}(\mathbb{K})$ be the Stiefel manifold with its volume measure $\operatorname{dVol}_{\operatorname{St}_{n,k}(\mathbb{K})}$. Let $X\operatorname{\overset{d}{=}}N(\mu,v^2I_{\operatorname{M}_{n,k}(\mathbb{K})})$ with $\mu\in\operatorname{St}_{n,k}(\mathbb{K})$, and write $p_{\operatorname{pr}(X)}$ for the density of its projection onto the Stiefel manifold. Small-variance conjecture. For small $v$, the intrinsic scalar variance satisfies

$$\frac{1}{\dim(\operatorname{St}_{n,k}(\mathbb{K}))}\int_{\operatorname{St}_{n,k}(\mathbb{K})}\operatorname{dist}^2(\mu,x)p_{\operatorname{pr}(X)}(x)\operatorname{dVol}_{\operatorname{St}_{n,k}(\mathbb{K})}(x)=v^2+\mathcal{O}(v^3).$$

This asymptotic would justify the two-point Padé approximation used for the intrinsic scalar variance of projected normal distributions on Stiefel manifolds; beyond the stated small-variance regime, the paper reports numerical evidence but does not establish the conjecture in general.