Shafarevich conjecture for projective hypersurfaces

Let $r\geq 2$ and $n\geq 1$. For a number field $K$ and a finite set $S$ of finite places of $K$, write $\mathcal{O}_{K,S}$ for the ring of $S$-integers. A smooth hypersurface of degree $r$ in $\mathbb{P}^{n+1}_{\mathcal{O}_{K,S}}$ is considered up to $\mathcal{O}_{K,S}$-linear isomorphism. Shafarevich conjecture for projective hypersurfaces. For all number fields $K$ and finite sets $S$ of finite places of $K$, the set of such isomorphism classes is finite. This is a finiteness conjecture for smooth hypersurfaces with prescribed degree and good reduction outside a fixed finite set; the supplied source does not indicate whether it has been resolved.