Completeness rigidity in dimensions greater than five

Determine whether every complete quasi-contact metric manifold of dimension greater than five is necessarily contact metric.

Background

The paper constructs strict quasi-contact metric manifolds in dimension five and, via products of cones, in every higher odd dimension. However, the higher-dimensional examples initially obtained are geodesically incomplete, and their sectional curvature becomes unbounded near a finite-distance end, preventing a smooth complete Riemannian extension. The authors therefore leave unresolved whether strict quasi-contact structures can exist in the complete setting in dimensions greater than five. A partial result under an additional nullity condition is cited, but the general question remains unanswered.

References

The analogy with nearly Sasakian geometry, nevertheless, suggests a question in the complete setting:

\emph{Is every complete quasi-contact metric manifold of dimension greater than $5$ necessarily contact metric?}

The above construction does not answer this question.

— Quasi-contact metric structures from Sasaki-Einstein metrics  (2610.02073 - Cappelletti-Montano et al., 1 Oct 2026) in Introduction, immediately following the construction of higher-dimensional examples; reiterated in Section 7, immediately after Proposition 7.2