KYY's multiplicative decomposition conjecture for shifted subgroups

Let $d\geq 2$ be fixed, let $q$ be a sufficiently large prime power with $q\equiv 1\pmod d$, and let $G\subseteq {\mathbb F}_q^*$ be the multiplicative subgroup of index $d$. For subsets $A,B\subseteq {\mathbb F}_q$, write $AB=\{ab:a\in A,\ b\in B\}$. KYY's conjecture. For every $\lambda\in {\mathbb F}_q^*$, the set $(G-\lambda)\setminus\{0\}$ has no nontrivial multiplicative decomposition; equivalently, there are no subsets $A,B$ with $|A|,|B|\geq 2$ such that

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

This is the proposed generalization of Sárközy's shifted-squares conjecture from fixed-index subgroups in finite fields; the supplied material gives no resolution, and the current paper's abstract instead presents the corresponding result as proved for every proper multiplicative subgroup.