C*-algebraic Sendov's conjecture

Let $\mathcal{A}$ be a unital C*-algebra, let $n\in\mathbb{N}\setminus\{1\}$, and let

$$p(z)=(z-a_1)(z-a_2)\cdots(z-a_n)\in\mathcal{A}[z]$$

with $a_1,\dots,a_n\in\overline{\mathbb{D}^*(0,1)}$. Define

$$p'(z)=\sum_{j=1}^n(z-a_1)\cdots\widehat{(z-a_j)}\cdots(z-a_n),\qquad z\in\mathcal{A},$$

where the hatted factor is omitted. C-algebraic Sendov's conjecture.* Assume that $p'$ admits roots $b_1,\dots,b_{n-1}\in\overline{\mathbb{D}^*(0,1)}$ in $\mathcal{A}$ and that each $b_k$ can be written in the form of the source's convexity equation. Then for every $a_j$, $1\leq j\leq n$, there exists a zero $b$ of $p'$ such that $b\in\overline{\mathbb{D}^*(a_j,1)}$. This extends the preceding commutative formulation to arbitrary C*-algebras without introducing differentiation; the source does not report a resolution, while its degree-two verification applies to the related formulation.