  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.
