ADE classification of Lagrangian fillings

Let $\Lambda\subseteq(\mathbb{R}^3,\xi_{\mathrm{st}})$ be the Legendrian rainbow closure of a positive braid whose mutable brick quiver is connected.

ADE classification of Lagrangian fillings. Exactly one of the following possibilities occurs:

1. $\Lambda$ is smoothly isotopic to the link of the $A_n$-singularity, and it has precisely $\frac{1}{n+2}\binom{2n+2}{n+1}$ exact Lagrangian fillings.
2. $\Lambda$ is smoothly isotopic to the link of the $D_n$-singularity, and it has precisely $\frac{3n-2}{n}\binom{2n-2}{n-1}$ exact Lagrangian fillings.
3. $\Lambda$ is smoothly isotopic to the link of the $E_6$, $E_7$, or $E_8$-singularity, and it has precisely $833$, $4160$, or $25080$ exact Lagrangian fillings, respectively.
4. $\Lambda$ has infinitely many exact Lagrangian fillings.

The conjecture proposes an ADE classification of the number of exact Lagrangian fillings for positive-braid rainbow closures with connected mutable brick quiver; the preceding discussion relates this expectation to cluster algebras and notes that it is not known for any Legendrian link in $(\mathbb{R}^3,\xi_{\mathrm{st}})$ except the standard Legendrian unknot.