Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers

Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted $k$-th powers. They conjectured that for each $k\ge 2$, if one changes $o(X^{1\mathrm{/}k})$ elements of $M_k^{\mathrm{\prime }}=\{x^k+1:x\in \mathbb{N}\}$ up to $X$, then the resulting set cannot be written as a product set $AB$ nontrivially. In this paper, we confirm a more general version of their conjecture for $k\ge 3$.