Conjecture on optimal distal cell-decomposition bounds in o-minimal structures

The discussion concerns an $o$-minimal structure, a finite family of formulas $\Phi(x;y)$, distal density, and distal cell decompositions; here $\abs{x}$ denotes the length of the object-variable tuple. The preceding results give bounds such as distal cell-decomposition exponent $2\abs{x}-4$ for $\abs{x}\geq 4$ over the ordered field $\mathbb{R}$.

Conjecture on optimal o-minimal bounds. The same bounds should hold in any $o$-minimal structure, potentially via Davenport–Schinzel sequences. Moreover, every $\Phi(x;y)$ in an $o$-minimal structure should have distal density $\abs{x}$, or even admit a distal cell decomposition of exponent exactly $\abs{x}$.

These claims would improve the general bounds discussed in the paper and would require new tools; the source presents them as conjectural extensions of the known results over $\mathbb{R}$.