Kurdyka's extension conjecture for arc-analytic semialgebraic functions

Let $X$ be an $\mathscr{A{\!}R}$-closed set in $\mathbb{R}^n$. Denote by $\mathscr{A}_a(X)$ the ring of arc-analytic semialgebraic functions on $X$, by $\mathscr{A}_a(\mathbb{R}^n)$ the corresponding ring on $\mathbb{R}^n$, and by $\mathcal{I}(X)$ the ideal of functions in $\mathscr{A}_a(\mathbb{R}^n)$ vanishing on $X$. Kurdyka's extension conjecture. Every arc-analytic semialgebraic function $f:X\to\mathbb{R}$ extends to an arc-analytic semialgebraic function on all of $\mathbb{R}^n$; equivalently, as $\mathbb{R}$-algebras,

$$\mathscr{A}_a(X)\simeq\mathscr{A}_a(\mathbb{R}^n)/\mathcal{I}(X).$$

The conjecture concerns the extension of arc-analytic functions from $\mathscr{A{\!}R}$-closed sets and the algebra–geometry correspondence in $\mathscr{A{\!}R}$ geometry. The paper presents it as an open question and does not provide evidence of a resolution.