Let $h$ be a function on the moduli space of abelian varieties of dimension $g$ over $\overline{\mathbf{Q}}$.

Let $K$ be a number field and let $g\geq 2$ be an integer. Fix a real number $C$. Does the finiteness of the set $$\{A/\overline{\mathbf{Q}}: \dim A = g, h(A) \leq C \}/\{\overline{\mathbf{Q}}-\mathrm{isomorphism}\}$$ imply the finiteness of the set $$\{A/K: \dim A=g, h(A_{\overline{\mathbf{Q}}}) \leq C, A/K \ \textrm{ has semi-stable reduction}\}/\{K-\mathrm{isomorphism}\}$$

In simple words, suppose you have a function on the moduli space of abelian varieties over $\overline{\mathbf{Q}}$ with the Northcott property. Then, can one deduce an **almost** Northcott property for abelian varieties over a fixed number field?