Assume that $G = \langle a, b \rangle$ is a finite non-abelian group which is generated by an involution $a$ and an element $b$ of order $n$ ($n\geq 3$) such that for every (complex) representation $\varphi$ of $G$ the matrix $\varphi(a) + \varphi(b) + \varphi(b^{-1})$ has only rational eigenvalues.

Question: Is there an upper bound on the order of $G$?