Annular link invariants from the Sarkar-Seed-Szab\'o spectral sequence

For a link in a thickened annulus $A\times I$, we define a $$\mathbb{Z}$ \oplus $\mathbb{Z}$ \oplus $\mathbb{Z}$$ filtration on Sarkar-Seed-Szab'o’s perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.