The majorization conjecture for twisted root maps

Let $\mathcal{C}_n$ be the class of polynomials of degree $n$ under consideration, let $\mathcal{A}'$ be the class of operators defined in the paper, and write $Z(P)$ for the multiset of zeros of a polynomial $P$. For a vector, $\Re$ and $\Im$ denote its componentwise real and imaginary parts, and $\prec$ denotes classical majorization.

Majorization conjecture. If $T\in\mathcal{A}'$ and $n\ge 1$, then for every $P\in\mathcal{C}_n$ at least one of the relations

$$\Re Z(P)\prec \Re Z\bigl(T(P)\bigr),\qquad \Im Z\bigl(T(P)\bigr)\prec \Im Z(P)$$

is valid.

The conjecture proposes a global alternative between majorization of the real parts in the forward direction and majorization of the imaginary parts in the reverse direction for the operators under study. It is motivated by numerical experiments and by local results for real parameters, while the preceding counterexamples show that analogous complex statements involving convex-hull inclusion or stronger forms of majorization cannot hold generally.