Faber–Voloch conjecture on Newton approximation

Let $g$ be a polynomial of degree $d\geq 2$ with coefficients in a number field $K$, and let $y_0\in K$. Define the Newton map

$$N(z)=z-\frac{g(z)}{g'(z)}$$

and, for each $n\geq 0$, set $y_{n+1}=N(y_n)$. Assume that the Newton approximation sequence $(y_n)$ is not eventually periodic. Let $C(K,g,y_0)$ be the set of primes $\fP$ of $K$ for which $(y_n)$ converges in the completion $K_{\fP}$ to a root of $f$. Faber–Voloch's conjecture. The natural density of the set $C(K,g,y_0)$ is zero. This conjecture predicts that, outside the eventually periodic case, Newton approximation converges $\fP$-adically to a root for a density-zero set of primes. The source presents it as a conjecture motivating the choice of polynomial, and no resolution is supplied here.