C*-algebraic Schoenberg conjecture

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

$$P(z)=(z-a_1)(z-a_2)\cdots(z-a_d)$$

be a polynomial over $\mathcal{A}$ with $a_1,\ldots,a_d\in\mathcal{A}$. Suppose that its formal derivative has the factorization

$$P'(z)=d(z-b_1)(z-b_2)\cdots(z-b_{d-1})$$

on $\mathcal{A}$, where $b_1,\ldots,b_{d-1}\in\mathcal{A}$. C-algebraic Schoenberg conjecture.* The inequalities

$$\sum_{k=1}^{d-1}b_kb_k^*\leq \frac{1}{d^2}\left(\sum_{j=1}^{d}a_j\right)\left(\sum_{j=1}^{d}a_j\right)^*+\frac{d-2}{d}\sum_{j=1}^{d}a_ja_j^*$$

and

$$\sum_{k=1}^{d-1}b_k^*b_k\leq \frac{1}{d^2}\left(\sum_{j=1}^{d}a_j\right)^*\left(\sum_{j=1}^{d}a_j\right)+\frac{d-2}{d}\sum_{j=1}^{d}a_j^*a_j$$

should hold. This extends the classical Schoenberg inequality from complex polynomials to C*-algebraic polynomials. The paper proves the conjecture for degree $2$; its general validity is left open.