The alpha-limit conjecture for generic real polynomial maps

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a generic polynomial map of degree $n\geq3$. A point of the Julia set $J_F$ of a rational map $F:\mathbb{R}^2\to\mathbb{R}^2$ is regular if some neighborhood $U$ has $J_F\cap U$ as a connected $1$-dimensional submanifold and contains points from two different basins. For a point $x$, let $\alpha_{N_f}(x)$ denote its alpha-limit set, and let $N_f$ be the Newton map of $f$.

Alpha-limit conjecture. There is some non-empty open subset $A\subset f(\mathbb{R}\mathrm{P}^2)$ such that $\alpha_{N_f}(x)$ is equal to the set of non-regular points of the boundary of $J_{N_f}$ for all $x\in A$.

This is one of the proposed two-dimensional analogues of Barna's simple-dynamics results for real Newton maps. The parser supplies no evidence that the conjecture has been resolved.