Finiteness conjecture for Fano threefolds with good reduction

Let $K$ be a number field and let $S$ be a finite set of finite places of $K$. A Fano threefold over $K$ has good reduction outside $S$ if it has good reduction at every finite place not in $S$.

Finiteness conjecture. The set of $K$-isomorphism classes of Fano threefolds over $K$ of Picard number $1$ with good reduction outside of $S$ is finite.

This question concerns arithmetic finiteness for Fano threefolds of Picard number $1.$The paper explains that existing methods do not yield new results for other such Fano threefolds, notably because infinitesimal Torelli may fail and the moduli may be non-separated but non-discrete. The status is not resolved in the supplied text.