The density-one conjecture for Newton approximation over global fields

Let $K$ be a finite extension of the rational field $\mathbb{Q}$ or of the rational function field $\mathbb{F}_p(T)$. Let $(x_n)$ be the Newton approximation sequence associated to a squarefree polynomial $f\in K[x]$ of degree at least $2$ and an initial point $x_0\in K$. Define $C(K,f,x_0)$ to be the set of places $v$ of $K$ for which $(x_n)$ fails to converge $v$-adically to some root of $f$. Newton approximation density-one conjecture. The set $C(K,f,x_0)$ has natural density one when ordered by norm. The conjecture extends the earlier number-field expectation to function fields. The paper proves only that the set of places where the Newton sequence diverges has positive lower density, so the density-one assertion remains open.