Unconditional implication from SH-fullness to superheaviness

Determine whether property (3) holds unconditionally: namely, whether every SH-full compact subset satisfying the stated relative symplectic cohomology condition is superheavy without separately assuming the equality of spectral invariants and a uniform bound on the Floer boundary depth.

Background

The paper recalls a criterion under which SH-fullness implies superheaviness. The criterion requires both an equality involving spectral invariants and a uniform boundary-depth bound. The authors state that they do not know whether this property holds without those additional assumptions.

References

Although we do not know whether property (3) holds unconditionally, the invariance $SH_M*(K)\cong SH_M*(\varphi(K))$ under symplectomorphisms $\varphi$ implies the following.

— Biran decomposition and relative symplectic cohomology  (2610.01334 - Jo et al., 1 Oct 2026) in Section 1, subsection “Heavy and superheavy subsets”