The squarefreeness conjecture for the polynomials $F_{m,n}$

Let $m,n\in\mathbb{N}_{+}$ with $n>2m$ and define

$$F_{m,n}(t)=A_{n-m}(1,t)^{2}-4A_{n-2m}(1,t)A_{n}(1,t).$$

Squarefreeness conjecture. The polynomial $F_{m,n}$ has no multiple roots.

This conjecture concerns the discriminant-type polynomials arising in the study of quadratic divisors of the quadrinomials. The source presents it as plausible but does not prove it.