Zero-set and extension conjecture for globally subanalytic arc-analytic functions

Let $E\subset\mathbb{R}^n$ be an $\mathscr{A{\\!}R}$-closed set. A function on $E$ is globally subanalytic arc-analytic when it is globally subanalytic and its composition with every analytic arc in $E$ is analytic. Zero-set and extension conjecture. Every $\mathscr{A{\\!}R}$-closed set is precisely the zero locus of a globally subanalytic arc-analytic function, and every globally subanalytic arc-analytic function on $E$ is the restriction to $E$ of a globally subanalytic arc-analytic function on $\mathbb{R}^n$. This conjecture seeks the globally subanalytic arc-symmetric analogue of the corresponding semialgebraic results; the supplied text gives no resolution, so its status remains open.