Genus-growth conjecture for geometric $p$-adic Lie towers

Let $U$ be a smooth curve over $k$ with compactification $C$. Let $M$ be an irreducible overconvergent $F$-isocrystal on $U$ whose fiber Newton polygon is constant as $x$ varies in $U$. Let $\eta$ be the smallest nonzero slope of $M$. The $p$-adic representation corresponding to the unit-root subcrystal of $M$ gives a $p$-adic Lie group $G$ of dimension $d$ and a $G$-tower of smooth curves

$$C=C_0\leftarrow C_1\leftarrow C_2\leftarrow\cdots.$$

Writing $g_n$ for the genus of $C_n$, there exist $m\in\mathbb{Z}$ with $\frac{m}{\eta}\in\mathbb{Z}$ and polynomials $a_0,\ldots,a_{m-1}\in\mathbb{Q}[x]$, each of degree $m(\frac{1}{\eta}+d)$, such that

$$g_{km+i}=a_i(p^k)$$

for $i=0,\ldots,m-1$ and $k\gg0$. Genus-growth conjecture. The genus sequence in this geometric $p$-adic Lie tower is eventually given on each residue class modulo $m$ by the stated polynomial in $p^k$. This predicts precise eventual quasipolynomial genus growth for towers arising from unit-root subcrystals. The conjecture is posed as a generalization of the paper's established genus-growth results, and its resolution is not supplied here.