On link quasimorphisms on the sphere and the equator conjecture

Link spectral invariants were introduced by Cristofaro-Gardiner, Humili\`ere, Mak, Seyfaddini, and Smith. They induce Hofer-Lipschitz quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-dimensional sphere. We prove that some linear combinations of those quasimorphisms vanish on the stabiliser of the equator. As a consequence, at least one of the following statements holds: there are non-trivial linear relations between the link quasimorphisms, or the space of equators of the sphere has infinite Hofer diameter. The proof relies on an `almost' K\"unneth formula in Link Floer Homology for some specific type of connected sums.