  We study a kinetically constrained Ising process (KCIP) associated with a
graph $G$ and density parameter $p$; this process is an interacting particle
system with state space $\{ 0, 1 \}^{G}$, the location of the particles. The
`constraint' in the name of the process refers to the rule that a vertex cannot
change its state unless it has at least one neighbour in state `1'. The KCIP
has been proposed by statistical physicists as a model for the glass
transition. In this note, we study the mixing time of a KCIP on the
2-dimensional torus $G = \mathbb{Z}_{L}^{2}$ in the low-density regime $p =
\frac{c}{L^{2}}$ for arbitrary $0 < c < \infty$, extending our previous results
for the analogous process on the torus $\mathbb{Z}_{L}^{d}$ in dimension $d
\geq 3$. Our general approach is similar, but the extension requires more
delicate bounds on the behaviour of the process at intermediate densities.
