Shiota's conjecture on Nash images of semialgebraic sets

Let ${\EuScript S}\subset{\mathbb R}^n$ be a semialgebraic set of dimension $d$. The set is pure dimensional if all its irreducible components have dimension $d$.

Shiota's conjecture. ${\EuScript S}$ is a Nash image of ${\mathbb R}^d$ if and only if ${\EuScript S}$ is pure dimensional and there exists an analytic path

$$\alpha:[0,1]\to{\EuScript S}$$

whose image meets all connected components of the set of regular points of ${\EuScript S}$.

This conjecture is intended to characterize precisely the semialgebraic sets that are Nash images of Euclidean space, providing an answer to the Nash-image variant of the stated problem. Its resolution is not established by the supplied context.