Characterization of $(\mathcal N,0)$-ats semialgebraic sets

A semialgebraic set $\EuScript T\subset\mathbb R^n$ is locally connected by analytic paths if, for every $x\in\EuScript T$ and every open semialgebraic neighborhood $U^x\subset\mathbb R^n$ of $x$, there is an open semialgebraic neighborhood $W^x\subset U^x$ of $x$ such that $\EuScript T\cap W^x$ is connected by analytic paths. The characterization conjecture. A semialgebraic set $\EuScript T\subset\mathbb R^n$ is an $((\mathcal N,0),\mathrm{ats})$ if and only if $\EuScript T$ is locally connected by analytic paths. The result is presented as a desired generalization of the paper's theorem for Nash manifolds with corners; the authors specifically hope to prove it when $\mu=0$, and no resolution is supplied here.