Supersingular reduction conjecture for non-isotrivial K3 surfaces

Let $k$ be an algebraically closed field of positive characteristic $p>0$, and let $C$ be a proper smooth curve over $k$ with function field $K=k(C)$. Let $X$ be a non-isotrivial, non-supersingular $K3$ surface over $K$. A semistable supersingular-reduction conjecture. After replacing $C$ by a finite covering, there exists a semistable family of combinatorial $K3$ surfaces $\mathscr{X} \to C$ extending $X$ such that

$$h(\mathscr{X}_v)>h(X)$$

for some closed point $v\in C$. This predicts a height jump, and hence supersingular reduction in the relevant cases, without restrictions on $p$, the degree of a quasi-polarization, or the generic height; the theorem proved in the paper establishes the corresponding result under additional assumptions.