Threshold dynamics for the 4$d$ mass-energy double critical NLS

We consider the 4$d$ mass-energy double critical NLS $$(i{\mathrm{\partial }}_t+\mathrm{\Delta })u=-\mathrm{\mid }u{\mathrm{\mid }}^{2}u+\mathrm{\mid }u\mathrm{\mid }u\mathrm{.}$$ In Luo (2024) and Cheng–Miao–Zhao (2016), the authors established a scattering/blowup dichotomy for solutions satisfying the energy constraint $E(u_0)<E^c(W)$, where $W$ is the energy-critical NLS ground state and $E^c$ is the energy for the underlying cubic NLS. We prove that the scattering/blowup dichotomy persists even at the energy threshold $E(u_0)=E^c(W)$.