The trace cone classifies Razak–Jacelon stabilizations

We prove that the canonical topological cone of all extended lower-semicontinuous tracial weights determines a separable nuclear C${}^{\ast }$-algebra after tensoring with the Razak–Jacelon algebra and the compact operators, answering Robert's trace-cone classification question positively. The isomorphism realizes the prescribed cone map, with arbitrary ideal structure and without a density assumption on the finite domains of the weights.