Marton's conjecture

Let $G$ be a group of bounded exponent, meaning that every element has order bounded by an absolute constant, and let $A\subset G$ satisfy $|A+A|\leq K|A|$. A subgroup cover is a subgroup $M$ whose translates cover $A$; the size of the cover is the number of translates, and $|M|$ denotes its cardinality.

Marton's conjecture. $A$ is $\exp(O(\log K))$-covered by a subgroup $M$ of size at most $\exp(O(\log K))|A|$.

This is the bounded-exponent formulation of the polynomial Freiman–Ruzsa conjecture and would improve the quasipolynomial covering and size bounds available in the theorem stated immediately beforehand.