3 problems

Cross-degree and interlacing conjecture. The inequalities

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

Conjecture on the solutions. All solutions of this equation, except possibly one and counting multiplicity, lie in the disk $|z|<\lambda_n$.