High-density ground-state and infinite-density ground-state existence conjecture

Let $u$ be a pair potential on ${\mathbb R}^d$. A ground-state configuration (GSC) is a configuration minimizing the energy per particle among the relevant configurations, and an infinite-density ground state (IDGS) in ${\mathbb R}^d$ is a nonzero positive Borel measure obtained as a vague limit of normalized Dirac combs of configurations whose densities and energy per particle diverge while $e_p(X_n)/\rho_n$ converges. The proposed limiting diagram is

$$\begin{CD} \text{GSC in }\Lambda @>\Lambda\to\infty>> \text{GSC in }{\mathbb R}^d\\ @V\rho\to\infty VV @VV\rho\to\infty V\\ \text{IDGS in }\Lambda @>>\Lambda\to\infty> \text{IDGS in }{\mathbb R}^d. \end{CD}$$

High-density ground-state and infinite-density ground-state conjecture. If $u$ is superstable and strongly tempered, then high-density GSCs and IDGSs in infinite space exist. Every IDGS in ${\mathbb R}^d$ is the vague limit both of periodic extensions of IDGSs in increasing volumes and of Dirac combs associated with GSCs of increasing density. Moreover, for every high-density GSC $X$, its associated measure $\mu_X$ is the vague limit of measures $\mu_{Y_n}$, where each $Y_n$ is the periodic extension of a ground state of $u_{\Lambda_n}$ in a parallelepiped $\Lambda_n$ and $\rho(Y_n)=\rho(X)$.

This claim describes the expected compatibility between the infinite-volume limit and the high-density limit for ground states and infinite-density ground states. The source presents it as an ultimate goal and gives no resolution, so its status remains open.