Sárközy's conjecture for shifted multiplicative subgroups

Let $d\geq2$, let $q$ be a prime power with $q\equiv1\pmod d$, and let $S_d=\{x^d:x\in\mathbb{F}_q^*\}$ be the subgroup of $d$-th powers. For subsets $A,B\subset\mathbb{F}_q^*$, write $AB=\{ab:a\in A,\ b\in B\}$; the decomposition is nontrivial when $|A|,|B|\geq2$.

Sárközy's conjecture. If $q$ is sufficiently large, then for every $\lambda\in\mathbb{F}_q^*$, the shifted subgroup $(S_d-\lambda)\setminus\{0\}$ has no nontrivial multiplicative decomposition: there do not exist $A,B\subset\mathbb{F}_q^*$ with $|A|,|B|\geq2$ such that

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

This generalizes the quadratic-residue case and concerns multiplicative structure in shifted finite-field subgroups. The paper reports progress toward the conjecture, while the supplied text gives no resolution.