Bremner and Ulas's reducibility-type conjecture for trinomials

Let $b,c\in\mathbb N$, and let

$$f(x)=x^n+b\epsilon_1x+c\epsilon_2$$

be a trinomial of degree $n\ge 4$, where $\epsilon_1,\epsilon_2\in\{\pm1\}$. A trinomial has reducibility type $(n_1,n_2,\ldots,n_k)$ if it is a product of irreducible polynomials in $\mathbb Q[x]$ with degrees $n_1,n_2,\ldots,n_k$.

Bremner and Ulas's conjecture. The reducibility type $(1,1,1,n-3)$ does not occur for $f(x)$.

The conjecture is part of the study of possible factor-degree patterns of trinomials. The source reports that the assertion remains a conjectural obstruction for degree $n\ge4$.