Conjectured improvements for concentration bounds in definable groups

Let $G$ be a definable group in an o-minimal structure, let $S\subset G$ be a definable subset avoiding unboundedly large arithmetic progressions, and let $k=\dim(S)$. Consider the random product $X=\prod_{i=1}^n g_i^{\pm1}$, where each $g_i\in G$ has order at least $s$ and the exponents are chosen independently and uniformly from $\{1,-1\}$. Write $\rho_S$ for the relevant concentration probability. Conjectured concentration improvements. Under the same assumptions, the following increasingly strong bounds are conjectured: (a) there exists $\epsilon>0$ depending on $S$ such that

$$\rho_S=O_S\mathopen{}\mathclose\bgroup\originalleft(\max(s^{-1},n^{-1/2})\aftergroup\egroup\originalright)^{(1/2^k)+\epsilon};$$

(b) there exists $\epsilon>0$ depending on $k$ and $G$ such that

$$\rho_S=O_S\mathopen{}\mathclose\bgroup\originalleft(\max(s^{-1},n^{-1/2})\aftergroup\egroup\originalright)^{(1/2^k)+\epsilon};$$

(c)

$$\rho_S=\max(s^{-1},n^{-1/2})^{1-o_S(1)};$$

(d)

$$\rho_S=O_S\mathopen{}\mathclose\bgroup\originalleft(\max(s^{-1},n^{-1/2})\aftergroup\egroup\originalright).$$

These conjectures propose that the preceding bound $O_S(\max(s^{-1/2^k},n^{-1/2^{k+1}}))$ is not tight. They form a hierarchy of progressively stronger improvements, with the final claim predicting the optimal-looking concentration estimate.