Conjecture on groups with exponentially few maximal sum-free sets

For an abelian group $G$, let $\mu(G)$ denote the size of a largest sum-free subset of $G$, and let $f_{\max}(G)$ denote the number of maximal sum-free subsets of $G$. Conjecture on an infinite class of groups. There exists a sequence of finite abelian groups $\{G_i\}_{i\in\mathbb{N}}$ of increasing order such that, for every $i$, $f_{\max}(G_i)$ is exponentially smaller than $2^{\mu(G_i)/2}$.

This conjecture asks for an infinite family where the previously proposed $2^{(1/2+o(1))\mu(G)}$ upper bound is not tight. The source says this was suspected in earlier work and gives no resolution here.