Non-vanishing conjecture for the polynomials $Q_{m,n}$ on the unit circle

Let $m>n$ be positive integers with $\gcd(m,n)=1$, and let $Q_{m,n}$ be the degree $2m-2n$ palindromic polynomial associated with this pair. Write $\mathbb D^*=\mathbb D\setminus\\{0\\}$ for the punctured unit disk. Non-vanishing conjecture. The polynomial $Q_{m,n}$ is non-vanishing on the unit circle. Consequently, $m-n$ of its roots lie inside $\mathbb D^*$. This conjecture concerns the general case of relatively prime positive integers with $m-n\geq 3$. The cases $m-n=3,4,6$ have been proved, while the case $m-n=5$ remains open; the general problem is still wide open, although infinitely many explicit pairs are known for each fixed difference.