Whitney's local holomorphic lamination conjecture for analytic varieties

Let $V$ be an analytic variety in $\mathbb C^n$ and let $X$ be a stratum of an analytic stratification of $V$. For every point $p\in X$, there should exist a neighborhood $U$ of $p$ in $V$, a metric space $T$, and a homeomorphism

$$\phi:(X\cap U)\times T\longrightarrow U$$

such that, for every $t\in T$, the restriction of $\phi$ to $(X\cap U)\times\{t\}$ is biholomorphic onto its image, the differentials of these restrictions are continuous on $(X\cap U)\times T$, and the lamination generated by $\phi^{-1}$, of the same dimension as $X$, is coherent with all the strata. Whitney's conjecture. Every analytic variety $V$ of $\mathbb C^n$ supports an analytic stratification with these properties. This is a local-triviality-type assertion for analytic stratifications, requiring holomorphic leaves and coherence with all strata; the source attributes the conjecture to H. Whitney (1965), while noting that Whitney's result gives a $b$-regular analytic stratification.