Stability threshold conjecture for ground states of the trapped nonlinear Schrödinger equation

Let $ngeq 1$, and let $p_n^*$ denote the upper exponent from the paper. For $+\frac{4}{n}<p<p_n^*$, let $\phi_\omega$ be the unique positive solution of

$$-\Delta \phi+|x|^2\phi+\omega\phi-\phi^p=0,\qquad x\in\mathbb{R}^n,$$

with $\phi_\omega(x)\to0$ as $|x|\to\infty$.

Stability threshold conjecture. For every $p$ satisfying $+\frac{4}{n}<p<p_n^*$, there exists $\omega_{p,n}$ such that $\phi_\omega$ is stable whenever $-n<\omega\leq\omega_{p,n}$ and unstable whenever $\omega>\omega_{p,n}$.

The question concerns the dynamical stability of the unique ground states in the $L^2$-supercritical range for the harmonic trapping potential. Stability is known near $\omega=-n$, while instability for sufficiently large $\omega$ is known; the conjecture asserts that these regimes are separated by a single threshold.