The Galois group conjecture for the polynomial Q_n

Let $n\geq 2$, let $Q_n$ be the polynomial studied in the paper, and write

$$D_{0,n}=\operatorname{Disc}(Q_n).$$

For a polynomial with rational coefficients, let $\operatorname{Gal}(Q_n)$ denote its Galois group and let $A_n$ and $S_n$ be the alternating and symmetric groups, respectively. Galois group conjecture. For integers $n\geq 2$,

$$\operatorname{Gal}(Q_n)=\begin{cases} A_n & \text{if $D_{0,n}$ is a square};\\ S_n & \text{if $D_{0,n}$ is not a square}. \end{cases}$$

The conjecture refines the general fact that a square discriminant forces the Galois group to be contained in an alternating group. It is supported by computations, but no resolution is given here.