Let $q$ be a formal variable and for every positive integer $n$ let $$[n]_q! = 1 (1 + q)(1 + q + q^2) \cdots (1 + q + \cdots + q^{n-1})$$ be the $q$-factorial.

Is it true that $[n!]_q + 1$ is an irreducible polynomial over $\mathbb{Z}$ for all positive integers $n$?

I already asked this question on MSE, and Sil verified it up to $n = 117$.


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