Genus stability conjecture for algebraic-geometric $\mathbb{Z}_p$-towers

Let $(C_n)_{n\geq 0}$ be a $\mathbb{Z}_p$-tower of curves over a finite field, with genus $g_n=g(C_n)$. Assume the tower comes from algebraic geometry. Genus stability conjecture. There are constants $a,b,c$ with $a>0$, depending on the tower, such that for all sufficiently large $n$,

$$g_n=ap^{2n}+bp^n+c.$$

This predicts a precise eventual form for the genus growth in geometrically arising towers; the supplied text gives no resolution status.