General $r$-paving bound via mixed characteristic polynomials

Let $M_n(\mathbb{C})^{+}_{1}$ denote the positive semidefinite $n\times n$ complex matrices bounded above by the identity, and let $\chi_r[A]$ be the $r$-characteristic polynomial of $A$. A partition $X_1\amalg\cdots\amalg X_r=[n]$ divides the index set $[n]=\{1,\ldots,n\}$ into $r$ parts. If $A\in M_n(\mathbb{C})^{+}_{1}$ has diagonal entries all at most $\delta\leq\left(\frac{r-1}{r}\right)^2$, then the general $r$-paving conjecture. there is a partition $X_1\amalg\cdots\amalg X_r=[n]$ such that

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

This is proposed as a plausible extension of the proved four-paving result, along the same lines; no resolution is given in the source.