Sárközy's multiplicative irreducibility conjecture for shifted quadratic residues

Let $p$ be a sufficiently large prime, and let $\mathcal R_p$ denote the set of nonzero squares in $\mathbb F_p$. For $\lambda\in\mathbb F_p^*$, consider the shifted set $(\mathcal R_p-\lambda)\setminus\{0\}$. Sárközy's multiplicative irreducibility conjecture. For every $\lambda\in\mathbb F_p^*$, the set $(\mathcal R_p-\lambda)\setminus\{0\}$ has no nontrivial multiplicative decomposition: there are no subsets $A,B\subseteq\mathbb F_p$ with $|A|,|B|\geq 2$ such that

$$AB=(\mathcal R_p-\lambda)\setminus\{0\}.$$

This is the multiplicative analogue of Sárközy's conjecture on additive irreducibility of the quadratic residues. The supplied context states that the conjecture was completely resolved by Kalmynin, so its database status is solved.