2-speciality conjecture for quadratic multiplier polynomials

For $m\geq 2$ and $n\geq 1$, let $P_{m,n}$ denote the multiplier polynomial in the quadratic case $d=2$. A polynomial is called 2-special when it has the special coefficient and divisibility properties defined earlier in the paper. 2-speciality conjecture. The polynomial $P_{m,n}$ is 2-special. The authors verified this computationally for all pairs $(m,n)$ with $m+n\leq 10$; by the paper's theorem on 2-special polynomials, it would imply the multiplier-resultant conjecture for $d=2$.