Homotopy CW-complex conjecture for ascending and descending links

In the situation of Theorem

, let $L^-_V(p)$ and $L^+_V(p)$ be the descending and ascending links, let ${\cal I}$ be the poset of closed unions of strata of ${\cal S}$, and let ${\cal L}^{\pm}_V(p)$ denote the corresponding sets of cells. **Homotopy CW-complex conjecture.** There exist CW-complexes $K^-_V(p)$ and $K^+_V(p)$, with cell sets ${\cal K}^-_V(p)$ and ${\cal K}^+_V(p)$, together with dimension-preserving bijections

k^-:{\cal K}^-_V(p)\to{\cal L}^-_V(p),\qquad k^+:{\cal K}^+_V(p)\to{\cal L}^+_V(p),

such that these bijections make the CW-complexes ${\cal I}$-filtered spaces and there are homotopy equivalences of ${\cal I}$-filtered spaces

h^-_V(p):K^-_V(p)\to L^-_V(p),\qquad h^+_V(p):K^+_V(p)\to L^+_V(p).

$$This is a homotopy-level refinement of the cellular description of the links in Theorem$$

, analogous to the classical Morse-theoretic result that a manifold is homotopy equivalent to a CW-complex whose cells correspond to critical points. The conjecture is presented as a strengthening of the earlier informal claim and is intended to provide filtered homotopy models for both links.