Complex spectral-order conjecture for polynomial zeros

Let $\mathcal{C}_{n}$ be the manifold of monic complex polynomials of degree $n$, and let $Z(P)$ denote the zero set of $P$, viewed as a subset of $\mathbb{R}^{2}$. For a set of complex zeros, write $\Re Z(P)$ for the real parts of its elements; let $D_{\lambda}$ be the differential operator used in the paper, and let $\preccurlyeq$ denote the induced spectral order.

Complex spectral-order conjecture. If $P\in\mathcal{C}_{n}$, then

$$\Re Z(P)\preccurlyeq \Re Z(D_{\lambda}P)$$

for any $\lambda\in\mathbb{R}$.

This is motivated by numerical calculations and by the corresponding order result for hyperbolic polynomials, while the preceding counterexamples show that a direct complex analogue with the spectral order on zero sets cannot hold in general. The conjecture is presented as an open question.