Small doubling implies small tripling for locally compact groups

Let $G$ be a locally compact group with Haar measure $\mu$, and let $S\subseteq G$ be a precompact symmetric open set containing the identity. Assume that, for some integer $n$ sufficiently large depending on $K$ and $\varepsilon$, one has

$$\mu(S^{\lceil(1+\varepsilon)n\rceil})\le K\mu(S^n).$$

Small tripling conjecture. For every $K\ge 1$ and $\varepsilon>0$, this implies

$$\mu(S^{3n})\ll_{K,\varepsilon}\mu(S^n).$$

This asks whether a weaker growth hypothesis than the usual tripling assumption already forces small tripling for sufficiently large radii. The supplied text presents it as a natural question and gives no resolution, so its status remains open.