Best-fit Stiefel tropical linear space at the Gaussian center

Let $X_1,\ldots,X_d$ be independent random variables with distribution $N(0,\sigma^2)$, and regard $(X_1,\ldots,X_d)$ as a point of $\mathbb{R}^{d+1}/\mathbb{R}\mathbf{1}$. Let $L_P$ be the Stiefel tropical linear space associated with $P$, and let the tropical distance be denoted by $d_{\rm tr}$.

Best-fit Stiefel tropical linear space conjecture. As $\sigma\to0$, the quantity

$$\frac{\mathbb{E}\left[d_{\rm tr}((X_1,\ldots,X_d),L_P)\right]}{\sigma}$$

achieves its minimum when $P=0$.

This is presented as the paper's ultimate goal for best-fitting non-hyperplane Stiefel tropical linear spaces. The supplied text gives no proof or resolution, so the conjecture remains open.