The real-part bound conjecture for non-real eigenvalues of ${\cal A}_{m;c}$

Let $m$ be a positive integer and let ${\cal A}_{m;c}$ denote the matrix pencil defined in the paper, with $c>0$. An eigenvalue is non-real when $\lambda\notin\mathbb{R}$. Real-part bound conjecture. If $\lambda$ is a non-real eigenvalue of ${\cal A}_{m;c}$, then $c<2$ and

$$|\lambda\pm c|<2,$$

and therefore

$$|\operatorname{Re}(\lambda)|\leq 2-c.$$

The conjecture would sharpen the known bound $|\operatorname{Re}(\lambda)|\leq |\lambda|<2+c$ for non-real eigenvalues. It is supported by numerical and asymptotic calculations for every $m$, but the source states that its proof appears substantially harder than expected and defers it to a separate paper.