Homotopy equivalence conjecture for ascending and descending links

Let $X$ be a complex stratified space with a stratification ${\cal S}$, let ${\cal I}$ be the set of all closed unions of strata of ${\cal S}$ ordered by inclusion, and let $L^\pm_V(p)$ be the ascending and descending links at a point $p$, with sets of cells ${\cal L}^\pm_V(p)$. Let $f$ be the stratified Morse function, let $E^\pm_f(p)$ be the corresponding upper and lower boundaries of the local Morse data, and let

$$l^\pm_V(p):L^\pm_V(p)\to E^\pm_f(p)$$

be the natural inclusion maps.

Homotopy equivalence conjecture. The maps $l^\pm_V(p)$ are homotopy equivalences of ${\cal I}$-filtered spaces.

This conjecture strengthens the preceding conjecture, which asserts that the ascending and descending links have filtered CW-complex models with corresponding cells and homotopy equivalences. It would identify the links directly with the boundary pieces of the local Morse data, relating the geometric ascending and descending sets to a central construction in stratified Morse theory. The supplied text gives no evidence of resolution.