Quantitative semi-algebraic basic lemma conjecture

Let $X \subset \mathbb{R}^k$ be a semi-algebraic set defined by a closed formula of complexity bounded by $C$, and let $\dim X=n$. A nested semi-algebraic filtration is a sequence of closed semi-algebraic subsets

$$Z^{(n-1)} \supset \cdots \supset Z^{(1)} \supset Z^{(0)}$$

with $\dim Z^{(i)}\leq i$ and $\operatorname{H}_q(X,Z^{(i)})=0$ for $0\leq q\leq i$. Quantitative semi-algebraic basic lemma conjecture. Such a filtration exists so that, for each $i$ with $0\leq i\leq n-1$, the complexity of $Z^{(i)}$ is bounded by $C^{O(k^{i+1})}$. Moreover, there is an algorithm computing closed formulas describing $Z^{(0)},\ldots,Z^{(\ell)}$, for $0\leq\ell\leq n-1$, whose complexity is bounded by $C^{k^{O(\ell)}}$. This would provide a potentially more efficient semi-algebraic replacement for the filtration obtained from skeleta of a triangulation, whose construction can have doubly exponential complexity; the conjecture remains open in the supplied text.