Irreducibility conjecture for trinomials with large middle coefficient

Let $a$, $m$ and $n$ be integers satisfying

$$n e 3,\qquad 0<m<n,\qquad \gcd(m,n)=1.$$

Irreducibility conjecture. If $|a|\geq 5$, then $x^{n}+ax^{m}\pm1$ is irreducible over $\mathbb{Q}$. Furthermore, there are only finitely many reducible polynomials of the form $x^{n}+ax^{m}\pm1$ with $|a|=3,4$; they are $x^{8}\pm3x^{3}-1$, $x^{8}\pm3x^{5}-1$, $x^{13}+3x^{4}-1$, $x^{13}-3x^{4}+1$, $x^{13}-3x^{6}-1$, $x^{13}+3x^{6}+1$, $x^{13}+3x^{7}\pm1$, $x^{13}-3x^{9}\pm1$, $x^{14}\pm4x^{5}-1$ and $x^{14}\pm4x^{9}-1$.

This conjecture would substantially strengthen the known irreducibility theorem, which assumes $|a|\geq n^{2}/3$, and is motivated by computational evidence and Perron’s earlier result.