The Gaussian minimal-energy asymptotics conjecture

Let $f(t)=e^{-\alpha t^2}$, with $0<\alpha<\pi e$. For configurations of density $\rho$ in $\mathbb{R}^n$, let the minimal $f$-energy be the infimum of the corresponding Gaussian pair energy. Gaussian minimal-energy asymptotics conjecture. The minimal $f$-energy is

$$(\rho+o(1))\left(\frac{\pi}{\alpha}\right)^{n/2}$$

as $n\to\infty$ with $\alpha$ and $\rho$ fixed, or more generally uniformly when $(\alpha,\rho)$ ranges over a compact subset of $(0,\pi e)\times(0,\infty)$. This would extend the sharp Gaussian energy bounds to the full range $\alpha<\pi e$ and improve the conditional expectation estimate for related inverse power-law potentials.