  We extend the recent work of Chong et al., (2022) to the critical case. More
precisely, we prove global in time, uniform in $N$ estimates for the solutions
$\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree--Fock--Bogoliubov
type with interaction potential $\frac1NV_N(x-y)=N^{2}v(N(x-y))$. We assume
that the potential $v$ is small which satisfies some technical conditions, and
the initial conditions have finite energy. The main ingredient is a sharp
estimate for the linear Schr\"odinger equation with potential in 6+1 dimension,
which may be of interest in its own right.
