In the following paper:
Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić, Real homotopy theory of semi-algebraic sets, Algebr. Geom. Topol. 11 (2011), no. 5, 2477–2545.
the authors construct a functor $\Omega^*_{PA}$ from the category of of semi-algebraic sets to the category of CDGAs, and prove that it is equivalent to the usual functor $A_{PL}(-;\mathbb{R})$ from rational (well, real here) homotopy theory, when restricted to compact semi-algebraic sets.
In Section 9.1, they explain that Kontsevich and Soibelmann, in a previous paper, suggested that this equivalence was true for any semi-algebraic set (and even for a bigger class of PA spaces), though the proof of Hardt–Lambrechts–Turchin–Volić does not work for noncompact sets. Has there been any progress on this conjecture? I could not find anything in the literature.
