The instability conjecture for the critical HLS stationary state

Let $\chi>0$, and let $\rho_{HLS,\lambda^*,c^*}\in{\mathcal Y}$ be the density whose parameters $(\lambda^*,c^*)$ are given by the stated parameter relation. Instability conjecture. The density $\rho_{HLS,\lambda^*,c^*}$ is an unstable stationary state of the equation. This expected instability is analogous to the instability described in the cited preceding work, but no proof or resolution is supplied here.