The conjecture on three rational linear factors of trinomials

Let $n\geq 4$, and consider trinomials $x^n+A x+B$ defined over $\mathbb{Q}$. Their reducibility type records the degrees of their irreducible factors over $\mathbb{Q}$; in particular, type $(1,1,1,n-3)$ means that they have three distinct rational roots and one remaining factor of degree $n-3$.

Three-linear-factor conjecture. There are no trinomials $x^n+A x+B$ defined over $\mathbb{Q}$ with reducibility type $(1,1,1,n-3)$.

For odd $n$, the claim is connected with the rational points on the curve defined by $G_n(p,q,r)=0$, whose genus is $(n-4)(n-3)/2$; the authors state that they do not know how to prove that its trivial points are complete. The even-$n$ case is ruled out by Descartes' rule of signs, so the conjectural content concerns the remaining cases.