Generalized Sárközy conjecture on additive decompositions of multiplicative subgroups

Let $d\geq 2$ be fixed, let $q$ be a prime power with $q\equiv 1\pmod d$, and let $G\subseteq {\mathbb F}_q^*$ be the multiplicative subgroup of index $d$. Generalized Sárközy conjecture. For all sufficiently large such $q$, the subgroup $G$ admits no nontrivial additive decomposition. The conjecture generalizes the finite-field square case from prime fields to multiplicative subgroups of fixed index; the source says it is widely believed and notes substantial progress over prime fields, while leaving the general formulation open.