Weak plainness conjecture for real Newton maps on the plane

Let $f:{\mathbb{R}^2}\to{\mathbb{R}^2}$ be a polynomial map of degree $n\geq3$ with $n$ distinct real roots $\{c_i\}$. Let $N_f$ be its Newton map, with Fatou set $F_{N_f}$, Julia set $J_{N_f}$, and basins ${\cal F}_{N_f}(c_i)$. A map is called weakly plain when it has the listed Julia-set, basin, and non-holomorphic behavior properties.

Weak plainness conjecture. Then:

1. $J_{N_f}$ is a countable union of wedge sums of a countable number of circles and of Cantor sets of circles of measure zero.
2. $F_{N_f}$ has no wandering domains.
3. The union of the basins ${\cal F}_{N_f}(c_i)$ has full Lebesgue measure.
4. Every neighborhood of any point of $J_{N_f}$ contains points from at least two distinct basins of attraction.
5. Unlike in the holomorphic case, basins of attraction are not necessarily simply connected, immediate basins of attraction are not necessarily unbounded, and $J_{N_f}$ can have interior points without being equal to the whole ${\mathbb{R}\mathrm{P}^2}$.

In particular, $N_f$ is weakly plain. The conjecture is a plane analogue of Barna's one-dimensional result and records several behaviors that distinguish real Newton maps from holomorphic maps. The source supplies numerical motivation but no resolution.