Hajdu–Sárközy's multiplicative irreducibility conjecture for perturbed shifted powers

Let $R$ be obtained from the set $\{x^k+1:x\in\mathbb{N}\}$ by changing $o(X^{1/k})$ elements up to $X$, where $k\geq 2$. A set $S\subset\mathbb{N}$ is multiplicatively irreducible if it cannot be written as a product set $AB=\{ab:a\in A,b\in B\}$ with $A,B\subset\mathbb{N}$ both having size at least $2$. Hajdu and Sárközy's conjecture. The new set $R$ is always multiplicatively irreducible. This is a multiplicative analogue of Erdős's conjecture on small perturbations of the squares. The paper's abstract says that a more general version is confirmed for $k\geq 3$, while the conjecture as stated here is not accompanied by a resolution for all $k\geq 2$.