Goresky–MacPherson cellular deformation-retraction conjecture

Let $X\cong\mathbb C^d$ be a complex vector space, let ${\cal S}$ be a complex analytic Whitney stratification of $X$, let $\mu$ be a Hermitian metric, let $p\in X$ be the origin, and let $f(x)=\operatorname{dist}_\mu(p,x)^2$ be Morse. For a regular value $a>0$, define

$$X^a=\{x\in X\mid f(x)\leq a\},\qquad \Sigma^a_f=\{p\in\Sigma_f\mid f(p)<a\}.$$

Goresky–MacPherson cellular deformation-retraction conjecture. There exists a $\nabla f$-like vector field $V$ on $X$ such that, with

$$X^a_c=\bigcup_{p\in\Sigma^a_f}M^-_V(p),$$

$X^a_c$ is a cellular subset of $X$ satisfying

$$\dim_{\mathbb R}(X^a_c\cap B)\leq\dim_{\mathbb C}B$$

for every $B\in{\cal S}$, and $X^a$ deformation retracts to $X^a_c$ by a stratum-preserving retraction. This is a variant of a conjecture of Goresky and MacPherson concerning cellular models for stratified spaces; the paper presents it as a potentially accessible direction for the constructed vector fields. Its resolution status is not specified in the supplied material.