Wan's genus-stability conjecture for p-adic Lie towers

Let $C$ be a smooth projective curve over a finite field of characteristic $p$, let $S$ be a finite set of closed points, and set $U=C\backslash S$. Let $\rho:\pi_1^{\mathrm{arith}}(U)\to \operatorname{GL}_r(\mathbb{Z}_p)$ be a $p$-adic Galois representation giving a $p$-adic Lie tower of smooth projective curves $\cdots\to C_1\to C_0\to C$. Write $g_i$ for the genus of $C_i$.

Wan's genus-stability conjecture. If $\rho$ is algebraic-geometric and ordinary, then the sequence $(g_i)$ is given by a polynomial in $p^i$ for $i\gg 0$.

This conjecture predicts stable polynomial growth of genera in certain geometric $p$-adic Lie towers. The paper studies the analogous characteristic-zero discriminant-growth statement, while the conjecture itself concerns towers of curves over finite fields of characteristic $p$.