Equidistribution of orbit sets realizing ECH spectral invariants

Determine whether, for a C∞-generic contact form λ, the orbit sets realizing the ECH spectral invariants converge after normalization as currents to the contact volume form, in the precise weak-current sense stated for C_k/√(2k).

Background

The paper discusses known equidistribution results for Reeb orbits and contrasts them with the stronger question of whether the particular orbit sets that realize ECH spectral invariants are themselves equidistributed.

The question is attributed to Irie and is presented as an unresolved problem. It concerns a contact 3-manifold (Y,ξ), a generic contact form λ, and orbit sets C_k associated to ECH spectral invariants in a fixed homology class and increasing relative grading.

References

For a $C\infty$-generic contact form $\lambda$, does $C_k/\sqrt{2k}$ weakly converge as a current to $\dd\lambda/\sqrt{vol(Y,\lambda)}$? I.e. for any one-form $\alpha$ is

— The ECH and alternative ECH capacities of closed symplectic 4-manifolds  (2608.19042 - Beiner, 19 Aug 2026) in Question 1.8, subsection “A question concerning equidistribution”