Barna-type plainness conjecture for real Newton maps

Let $p$ be a generic real polynomial of degree $n\geq4$ without complex roots, with roots $r_1,\dots,r_n$, and let $N_p$ be its Newton map. A polynomial is called plain when its Newton map has the dynamical properties described by Barna's theorem.

Plainness conjecture. Every $p$ satisfying the hypotheses of Barna's theorem is a plain map.

This conjecture proposes that the full-measure basin and Julia-set behavior established by Barna for generic real polynomials without complex roots extends to the corresponding class of real Newton maps.