Robert's trace-cone question: does the cone of extended traces classify W-stabilized separable nuclear C*-algebras?

Theorem 1.1: if $A,B$ are separable nuclear complex C*-algebras and $T(A)\cong T(B)$ as topological cones, then $A\otimes\mathcal W\otimes\mathcal K\cong B\otimes\mathcal W\otimes\mathcal K$, and the isomorphism can be chosen to induce the given cone map. No density condition on finite domains of the weights and no UCT is assumed. It does not classify algebras without the $\mathcal W\otimes\mathcal K$ stabilization, and does not determine the range of the invariant.

Theorem 1.1: if $A,B$ are separable nuclear complex C*-algebras and $T(A)\cong T(B)$ as topological cones, then $A\otimes\mathcal W\otimes\mathcal K\cong B\otimes\mathcal W\otimes\mathcal K$, and the isomorphism can be chosen to induce the given cone map. No density condition on finite domains of the weights and no UCT is assumed. It does not classify algebras without the $\mathcal W\otimes\mathcal K$ stabilization, and does not determine the range of the invariant.