Nash–Semple conjecture on resolution by Nash transformations

Let $X$ be a pure $d$-dimensional $\mathbb{K}$-analytic set, where $\mathbb{K}$ is $\mathbb{C}$ or $\mathbb{R}$, and let $\mathcal{N}(X)$ denote its Nash transformation. A Nash–Semple conjecture asserts that a finite succession of Nash transformations resolves the singularities of $X$. Nash proposed this problem in a private communication to Hironaka, and Semple posed it independently. Spivakovsky gave a relevant partial answer, but the conjecture is disproved by counterexamples in every dimension $d\geq 4$.