On the properness of the moduli space of stable surfaces over $\mathbb{Z}[1\mathrm{/}30]$

We show the properness of the moduli stack of stable surfaces over $\mathbb{Z}[1\mathrm{/}30]$, assuming the locally-stable reduction conjecture for stable surfaces. This relies on a local Kawamata–Viehweg vanishing theorem for for 3-dimensional log canonical singularities at closed point of characteristic $p\mathrm{\ne }2,3$ and $5$ which are not log canonical centres.