4 problems

Simple-roots conjecture. For almost all polynomials $p$, the polynomial $\Delta_{\theta,h}(p)$ has only simple roots, apart from the stated exclusion.

Self-interlacing conjecture. The polynomial $\Delta_{\theta,h}(p)(x)$ is self-interlacing, or is a self-interlacing polynomial multiplied by $x$.

Finite-difference stability conjecture. For any stable polynomial $p$, the polynomials

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