Nonexistence conjecture for trace polynomials in type $(2,4,2)$

Let $m$ be an odd integer with $m\geq 5$, and let $W\in\mathbb{Z}_2*\mathbb{Z}_4$. Let $\tau_W(\lambda)$ denote the trace polynomial associated with $W$. The trace-polynomial nonexistence conjecture. There is no word $W\in\mathbb{Z}_2*\mathbb{Z}_4$ for which

$$\tau_W(\lambda)=\sqrt{2}(\lambda^2-1)^m.$$

The conjecture is supported by computational evidence and, according to the source, would imply the Rosenberger conjecture for generalised triangle groups of type $(2,4,2)$. It remains open.