The Rasmussen–Tamagawa finiteness conjecture for heavenly abelian varieties

Let $K$ be a number field and let $g\geq 0$. For a rational prime $\ell$, let $\mathcal{H}(K,g,\ell)$ be the set of $K$-isomorphism classes of $g$-dimensional abelian varieties that are heavenly at $\ell$, and define

$$\mathcal{H}(K,g):=\bigl\{\bigl([A]_K,\ell\bigr):[A]_K\in\mathcal{H}(K,g,\ell)\bigr\}.$$

Rasmussen–Tamagawa conjecture. The set $\mathcal{H}(K,g)$ is finite for any choice of $K$ and $g$. This strengthens the known finiteness of $\mathcal{H}(K,g,\ell)$ when the prime $\ell$ is fixed; it remains open, although substantial progress is known.