Lagrangian fillings for Legendrian links of finite or affine Dynkin type

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type ADE or affine type DE. We also provide as many Lagrangian fillings with rotational symmetry as seeds of type B, G₂, G₂, B, or C₂, and with conjugation symmetry as seeds of type F₄, C, E⁽²⁾ ₆, F₄, or A⁽²⁾₅. These families are the first known Legendrian links with (infinitely many) exact Lagrangian fillings (with symmetry) that exhaust all seeds in the corresponding cluster structures beyond type AD. Furthermore, we show that the N-graph realization of (twice of) Coxeter mutation of type DE corresponds to a Legendrian loop of the corresponding Legendrian links.