General maximum-root bound for the mixed characteristic polynomial

Let $M_n(\mathbb{C})^{+}$ denote the positive semidefinite $n\times n$ complex matrices, and let a positive contraction be a positive semidefinite matrix bounded above by the identity. For $r\in\mathbb{N}\setminus\{1\}$, let $\chi_r[A]$ denote the $r$-characteristic polynomial of $A$, and let $\operatorname{max root}\chi_r[A]$ be its largest root. If $A\in M_n(\mathbb{C})^{+}$ is a positive contraction whose diagonal entries are all at most $\delta$, then the mixed-determinant root bound conjecture.

$$\operatorname{max root}\chi_r[A]\leq \frac{1}{r}\left(\sqrt{1-\delta}+\sqrt{(r-1)\delta}\right)^2.$$

The text presents this as a plausible estimate for general $r$, following proved estimates for $r=2,3,4$; its resolution is not supplied here.