Monotonicity of Gaussian tropical distance to the diagonal hyperplane

Let $X_1,\ldots,X_k$ be independent Gaussian random variables with distribution $N(0,\sigma^2)$, and let $H_0$ denote the tropical hyperplane whose relevant coordinates are equal. The tropical distance is denoted by $d_{\rm tr}$.

$$\mathbb{E}\left[d_{\rm tr}((X_1,\ldots,X_k),H_0)\right]$$

Monotonicity conjecture. The expectation above monotonically decreases with $k$. Consequently, as $\sigma\to0$, the hyperplane that best fits $X$ converges to $H_0$, meaning that the apex is at the center of the Gaussian.

The claim would identify the apex-centered hyperplane as the best-fitting hyperplane in the small-variance limit for general dimension. The surrounding discussion gives numerical evidence and notes that the cases $k=2,3$ agree with established results, but no resolution of the general monotonicity statement is supplied.