The reducibility conjecture for falling-factorial polynomials

For integers $a\geq k+3$ and $k\geq1$, define the falling factorial

$$x^{\underline{k}}=x(x-1)\cdots(x-k+1).$$

Reducibility conjecture. If $x^{\underline{k}}-a!$ is reducible over $\mathbb{Z}$, then $(k,a)=(3,6)$, $(k,a)=(4,7)$, or there exists an integer $t$ such that $a=t!-1$ and $k=t!-t$. The source presents this as a conjecture motivated by numerical evidence. The exceptional pairs correspond to the sporadic nontrivial solution $(6,7,10)$, while the family $a=t!-1$, $k=t!-t$ corresponds to class $1$ solutions; the general reducibility question remains open.