Sharpness of the positive-first-Chern-class criterion

Prove that if a closed symplectic 4-manifold (X,Ω) satisfies c_1(TX,Ω)·[Ω]>0, then every four-dimensional Liouville domain symplectically embedded in (X,Ω) has finite ECH capacities in every degree.

Background

The paper proves that a closed rational symplectic 4-manifold satisfying c_1(TX,Ω)·[Ω]≤0 contains an embedded Liouville subdomain with infinite ECH capacities. The conjecture asserts that this condition is sharp: when the pairing is positive, no embedded Liouville domain should have infinite ECH capacities.

The conjecture is established in the paper for Weinstein domains using work of Mark–Tosun and Chen. Extending the argument from Weinstein domains to arbitrary Liouville domains would settle the stated conjecture.