Local ramification-growth conjecture for unit-root $F$-isocrystals

Let $M$ be an irreducible overconvergent $F$-isocrystal on $\operatorname{Spec}(F)$ with nonnegative slopes. Let $M^{u-r}$ be its unit-root subcrystal, and let $\eta$ be its first nonzero slope. Consider the corresponding $p$-adic Lie extension $F_\infty/F$ and write $s_n$ for the largest ramification break of $F_n/F$. Local ramification-growth conjecture. There exist $m\in\mathbb{Z}$ with $\frac{m}{\eta}\in\mathbb{Z}$ and constants $a_0,\ldots,a_{m-1},b_0,\ldots,b_{m-1}\in\mathbb{Q}$ such that

$$s_{km+i}=a_i p^{mrk}+b_i$$

for $i=0,\ldots,m-1$ and $k\gg0$. This is the local counterpart to the global genus-growth prediction: the first nonzero Frobenius slope is expected to control eventual ramification growth in the associated $p$-adic Lie extension. The source gives no resolution of this local conjecture.