C*-algebraic de Bruin-Sharma 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

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

on $\mathcal{A}$, where $b_1,\ldots,b_{d-1}\in\mathcal{A}$. C-algebraic de Bruin-Sharma conjecture.* If

$$\sum_{j=1}^{d}a_j=0,$$

then

$$\sum_{k=1}^{d-1}(b_kb_k^*)^2\leq \frac{2}{d^2}\left(\sum_{j=1}^{d}a_ja_j^*\right)^2+\frac{d-4}{d}\sum_{j=1}^{d}(a_ja_j^*)^2$$

and

$$\sum_{k=1}^{d-1}(b_k^*b_k)^2\leq \frac{2}{d^2}\left(\sum_{j=1}^{d}a_j^*a_j\right)^2+\frac{d-4}{d}\sum_{j=1}^{d}(a_j^*a_j)^2.$$

This is the C*-algebraic higher-order analogue of the classical de Bruin-Sharma inequality. The source states the degree-$2$ case is clear, but gives no resolution of the general C*-algebraic conjecture.