Let $A\in\mathbb{R}^{2\times 2}$ and assume that $|{\rm tr}A|<4$. Prove or disprove that there exist $a_1,a_2,a_3,a_4\in\mathbb{C}$ such that $|a_1|<1$, $|a_2|<1$, $|a_3|<1$, $|a_4|<1$, $p(x)\triangleq\prod_{i=1}^{4} (x-a_i)$ is a polynomial with real coefficients, and
\begin{equation} b_1I_2+b_2A+b_3A^2+A^3=0, \end{equation}
where for all $i\in\{1,2,3\}$, \begin{gather} b_i\triangleq(-1)^i\mspace{-40mu}\sum_{j_1,\ldots,j_{4-i}\in\{1,\ldots,4\}}\mspace{-40mu}a_{j_1}\times \cdots\times a_{j_{4-i}}, \end{gather} that is,
\begin{align} b_1&\triangleq-a_1a_2a_3-a_1a_2a_4-a_2a_3a_4-a_1a_3a_4,\\ b_2&\triangleq a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4,\\ b_3&\triangleq -a_1-a_2-a_3-a_4. \end{align}