The NE/EN vertex conjecture for inner polynomials

Let $Z$ be the innermost polynomial and $\zeta(F)$ the inner polynomial arising from the paper's construction. For a convex polygon, the NE vertex is the uppermost vertex with largest $x$-coordinate among ties, and the EN vertex is the rightmost vertex with largest $y$-coordinate among ties. The NE/EN vertex conjecture. The Newton polygon of $Z$ has an NE vertex $(m',n')$ coinciding with its EN vertex, satisfying

$$\frac{n'}{m'}=\frac{n}{m},\qquad N^0(Z)\subseteq\operatorname{Rect}_{m',n'},$$

and the Newton polygon of $\zeta(F)$ has the same property. The paper identifies this as an important step toward proving its preceding conjecture; no resolution is supplied.