Countably many isogeny classes for every attainable Picard number

Let $R_g^{(p)}$ denote the set of Picard numbers of $g$-dimensional abelian varieties in characteristic $p>0$. Let $g \geq 1$ and let $\rho \in R_g^{(p)}$. Countable-isogeny-class conjecture. There exist countably many isogeny classes of abelian varieties of dimension $g$ and Picard number $\rho$.

The source says that this conjecture would imply the preceding recursive description of Picard numbers. Further results are surveyed by Oort, including known cases and open questions, so the conjecture remains open.