Counterexample to positively curved Hopf.

Y'all know the drill. The arxiv link is arxiv.org/abs/2609.11980 It is a positively curved metric S^3xS^3, hence a positively curved 6-manifold whose euler characteristic is not strictly positive. The obvious question is whether there is also a counter to negative Hopf (a closed, negatively curved 6 manifold whose euler is not strictly negative), but the methods of the current paper don't seem to help with that.