Integral-coefficient spectral diameter conjecture

Establish that, for every closed symplectic manifold $(M,\\omega)$, the spectral diameter of $Ham(M,\\omega)$ is infinite when the spectral metric is defined using $\\mathbb Z$-coefficients.

Background

The authors note that the spectral metric depends on the coefficient ring. In particular, finite spectral diameter over field coefficients does not preclude infinite diameter over the integers, motivating the stronger integral-coefficient conjecture.

References

In light of this development, one could ambitiously conjecture that, for every closed symplectic manifold $(M,\omega)$, the spectral diameter of $Ham(M,\omega)$ is infinite over $Z$.

— Biran decomposition and relative symplectic cohomology  (2610.01334 - Jo et al., 1 Oct 2026) in Section 1, Introduction

It seems that it is unknown whether the spectral diameter of $Ham(\mathbb{CP}1,\omega_\mathrm{FS})$ is infinite.

— Biran decomposition and relative symplectic cohomology  (2610.01334 - Jo et al., 1 Oct 2026) in Section 1, subsection “The large scale geometry of $Ham(M,\\omega)$”