Splitting-field growth conjecture for zeta functions of $\mathbb{Z}_p$-towers

Let $P(C_n,s)$ be the numerator of the zeta function of the curve $C_n$. Let $\mathbb{Q}_n$ and $\mathbb{Q}_{p,n}$ be its splitting fields over $\mathbb{Q}$ and $\mathbb{Q}_p$, respectively, and write $[\mathbb{Q}_{p,n}:\mathbb{Q}_p]^{\rm ram}$ for the ramification degree. Splitting-field growth conjecture. The following assertions hold as $n\to\infty$:

1. $[\mathbb{Q}_n:\mathbb{Q}]\to\infty$.
2. $[\mathbb{Q}_{p,n}:\mathbb{Q}_p]\to\infty$.
3. $[\mathbb{Q}_{p,n}:\mathbb{Q}_p]^{\rm ram}\to\infty$.
4. There is a positive constant $c$, depending on the tower, such that for all sufficiently large $n$,

$$[\mathbb{Q}_{p,n}:\mathbb{Q}_p]^{\rm ram}\geq cp^n.$$

The conjecture concerns the arithmetic and local complexity of the Frobenius-root fields in a $\mathbb{Z}_p$-tower; the supplied text gives no resolution status.