Multiplicative irreducibility of shifted multiplicative subgroups in the extremal case

In a recent breakthrough, Kalmynin proved a conjecture of S\'ark\"ozy on additive irreducibility of the set of quadratic residues in a prime field. More recently, Kim, Yip, and Yoo initiated the study of a multiplicative analogue of the conjecture for shifted multiplicative subgroups. Specifically, they showed that for an odd prime $p$, a proper multiplicative subgroup $G$ of $\mathbb F_p^*$, and $\lambda\in G$, there do not exist sets $A,B\subseteq \mathbb F_p^*$ with $|A|,|B|\ge 2$ such that $AB=(G-\lambda)\setminus\{0\}$. In this paper, when $\lambda \in \mathbb F_p^* \setminus G$, we completely resolve this problem in the equality case from a Stepanov bound in a prime field.