Conjecture on principally polarized varieties in almost ordinary isogeny classes

Let $d54a_h$ denote an almost ordinary isogeny class defined over $d53d_q$. For each positive integer $n$, consider the principally polarized $d53d_{q^n}$-rational abelian varieties in $d54a_h$.

Principally polarized isogeny-class size conjecture. For a positive proportion of integers $n$, their number is

$$q^{n(\dim \mathcal{A}_g -1)(1/2 + o(1))}.$$

Furthermore, this quantity is an upper bound for the number for all $n$.

This conjecture predicts the asymptotic size of principally polarized members of an almost ordinary isogeny class over finite extensions, and an upper bound valid for every extension degree. The supplied text gives no resolution status.