Gontier–Lewin–Nazar's finite-rank optimizer conjecture

Let $d\geq 2$, let $q>1$, and let $\gamma$ be a density matrix satisfying $0\leq\gamma\leq 1$. Write

$$\gamma=\sum_{n=1}^N |u_n\rangle\langle u_n|$$

for a rank-$N$ projection, and let $\rho_\gamma$ denote its density. The associated ground-state equations are

$$(-\Delta-\rho_\gamma^{q-1})u_n=-\mu_nu_n,$$

where $-\mu_1< -\mu_2<\cdots\leq -\mu_N\leq 0$ are the $N$ lowest eigenvalues of $-\Delta-\rho_\gamma^{q-1}$ on $L^2(\mathbb{R}^d)$. Optimizers of Lieb--Thirring interpolation inequalities. In the mass-subcritical case $q<1+2/d$, for every $N\geq 2$ there exists a ground-state solution of rank $N$; in the mass-supercritical case $q>1+2/d$, there exists a finite integer $N_0$ such that no ground state of rank $N$ exists for $N>N_0$. In particular, there exists a global optimizer of finite rank. The conjecture formulates a sharp dichotomy between the mass-subcritical and mass-supercritical regimes. The subcritical assertion was previously proved only for $q$ sufficiently close to $1$, with partial results across the full range, while the stated supercritical finite-rank conclusion remained open in the source.