Necessity of the no-conjugate-points hypothesis

Determine whether the absence of conjugate points is necessary for unboundedness of the Lagrangian spectral distance under the remaining hypotheses of Theorem 1.1, namely nontrivial first cohomology and nonvanishing Euler characteristic.

Background

The main unboundedness theorem assumes that the closed connected manifold admits a Riemannian metric without conjugate points, has nontrivial first cohomology with coefficients in the chosen ring, and has Euler characteristic nonzero in that ring.

The proof uses the no-conjugate-points assumption to construct a global generating function on the tangent bundle via the exponential map. The authors explain that an alternative Floer-theoretic approach avoids this assumption for part of the argument, but that their proof of vanishing of the fundamental-class spectral invariant still requires a unique minimizing geodesic construction, which is guaranteed by the absence of conjugate points. They explicitly leave unresolved whether this geometric hypothesis is genuinely needed for the theorem.