  We consider two sharp next-order asymptotics problems, namely the asymptotics
for the minimum energy for optimal point configurations and the asymptotics for
the many-marginals Optimal Transport, in both cases with Riesz costs with
inverse power-law long range interactions. The first problem describes the
ground state of a Coulomb or Riesz gas, while the second appears as a
semiclassical limit of the Density Functional Theory energy modelling a quantum
version of the same system. Recently the second-order term in these expansions
was precisely described, and corresponds respectively to a Jellium and to a
Uniform Electron Gas model.
  The present work shows that for inverse-power-law interactions with power
$s\in [d-2,d)$ in $d$ dimensions, the two problems have the same minimum value
asymptotically. For the Coulomb case in $d=3$, our result verifies the
physicists' long-standing conjecture regarding the equality of the second-order
terms for these two problems. Furthermore, our work implies that, whereas
minimum values are equal, the minimizers may be different. Moreover, provided
that the crystallization hypothesis in $d=3$ holds, which is analogous to
Abrikosov's conjecture in $d=2$, then our result verifies the physicists'
conjectured $\approx 1.4442$ lower bound on the famous Lieb-Oxford constant.
Our work also rigorously confirms some of the predictions formulated by
physicists, regarding the optimal value of the Uniform Electron Gas
second-order asymptotic term.
  We also show that on the whole range $s\in(0,d)$, the Uniform Electron Gas
second-order constant is continuous in $s$.
