The approximate polynomial growth conjecture for locally compact groups

Let $G$ be a locally compact group, let $A\subset G$ be a compact neighborhood, and let $\mu$ be Haar measure on $G$. For $n\geqslant 1$, write $A^n$ for the $n$-fold product set of $A$. Approximate polynomial growth conjecture. If

$$\mu(A^n)\leqslant n^d\mu(A)$$

for all $n\geqslant d\log d$, then $A$ is contained in a $d^{1+o(1)}$-dimensional ball $B$ of some continuous translation-invariant pseudometric, with

$$\mu(B)\leqslant \exp(d^{1+o(1)})\mu(A).$$

The source proposes this as a non-abelian analogue of its weak Freiman theorem, explicitly describing it as a conjectural direction and giving no resolution.