Algebraization conjecture for compact Nash sets

A compact Nash set is a compact semialgebraic set endowed with its Nash structure. Two compact Nash sets are strongly asym-Nash cobordant if they are related by the strong form of asymmetric Nash cobordism, and they are asym-Nash cobordant if they are related by asymmetric Nash cobordism. Algebraization conjecture. Every compact Nash set is strongly asym-Nash cobordant to a compact real algebraic set; equivalently, every compact Nash set is asym-Nash cobordant to a point. This conjecture asks whether asymmetric Nash cobordism provides an algebraic representative for every compact Nash set. The supplied text gives no evidence that the conjecture has been resolved.