Commutative C*-algebraic Sendov's conjecture

Let $\mathcal{A}$ be a unital commutative C*-algebra whose group of invertible elements $G(\mathcal{A})$ is dense in $\mathcal{A}$. Let $n\in\mathbb{N}\setminus\{1\}$ and

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

where $a_1,\dots,a_n\in\overline{\mathbb{D}^*(0,1)}$. The C*-algebraic closed disc is defined by

$$\overline{\mathbb{D}^*(a,r)}=\{z\in\mathcal{A}:(z-a)(z-a)^*\leq\sqrt r\cdot1\}.$$

Commutative 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

$$b_k=\sum_{j=1}^n\omega_{b_{k,j}}a_j,\qquad \sum_{j=1}^n\omega_{b_{k,j}}=1,$$

with positive coefficients as in the source's convexity equation. Then for each $a_j$, $1\leq j\leq n$, there exists a zero $b$ of $p'$ such that $b\in\overline{\mathbb{D}^*(a_j,1)}$. This is the commutative C*-algebraic formulation introduced by the paper; the source verifies it only in the degree-two case, so the general statement remains open.