The zero-strip-decreasing conjecture for complex finite difference operators

Let $\mathbb{C}[z]$ denote the ring of complex polynomials, and let $\Delta_{\theta,h}$ be the finite difference operator defined in the paper. An operator is zero strip decreasing when it does not increase the width of the smallest horizontal strip containing the zeros of a polynomial.

Zero-strip-decreasing conjecture. The operator $\Delta_{\theta,h}$ defined on $\mathbb{C}[z]$ is zero strip decreasing.

The corresponding property was proved in the cited work for the operator acting on $\mathbb{R}[z]$; the source reports calculations suggesting that it extends to arbitrary complex polynomials, but gives no proof or resolution.