Ruzsa's conjecture on spanning size in abelian groups of finite torsion

Let $G$ be an abelian group with finite torsion $r$, and let $A$ be a finite subset of $G$. Write $A+A=\{a+b\mid a,b\in A\}$ and let $\langle A\rangle$ be the smallest subgroup or coset of a subgroup containing $A$. The doubling constant is $|A+A|/|A|$.

Ruzsa's conjecture. If there exists a constant $K$ such that

$$\frac{|A+A|}{|A|}\leq K,$$

then there exists a constant $C\geq 2$ such that

$$\frac{|\langle A\rangle|}{|A|}\leq r^{CK}.$$

This conjecture predicts that the exponential dependence on the doubling parameter in the Freiman--Ruzsa theorem is essentially optimal, while replacing the theorem's much larger exponent by a linear function of $K$. The supplied text gives no resolution status.