  We investigate positive braid Legendrian links via a Floer-theoretic approach
and prove that their augmentation varieties are cluster K2 (aka. A-) varieties.
Using the exact Lagrangian cobordisms of Legendrian links in [EHK16], we prove
that a large family of exact Lagrangian fillings of positive braid Legendrian
links correspond to cluster seeds of their augmentation varieties. We solve the
infinite-filling problem for positive braid Legendrian links; i.e., whenever a
positive braid Legendrian link is not of type ADE, it admits infinitely many
exact Lagrangian fillings up to Hamiltonian isotopy.
