Refuting the X-MEMS closest-separable-state conjecture in arXiv:2209.01430v2, Appendix A, Eq. (A11)

We refute the closest-fully-separable-state shape conjecture of Consiglio, Apollaro and Wieśniak, arXiv:2209.01430v2, Appendix A, Eq. (A11), already for three qubits on the low branch at gamma = 11/112. Every fully separable state of the proposed shape has squared Hilbert–Schmidt distance at least 109/23520, whereas an explicit mixture of ten pure product states has distance 417965512477/90316800000000. The counterexample and exact restricted minimum have self-contained written proofs, as do related counterexamples to the full-separability and concurrence comparisons in the posing source. An elementary bound also establishes its large-qubit limit without the conjectured shape.