BCFG classification of symmetric Lagrangian fillings

Let $\Lambda(\beta)\subseteq(\mathbb{S}^3,\xi_{\mathrm{st}})$ be the Legendrian rainbow closure of a positive braid $\beta$. An exact Lagrangian $G$-filling is an exact Lagrangian filling invariant under a faithful finite-group action $G$ by exact symplectomorphisms, with $G(\Lambda)=\Lambda$ setwise.

BCFG classification of Lagrangian fillings. The following symmetric cases have the stated numbers of fillings:

1. If $\Lambda(\beta)=\Lambda(A_{2n-1})$, the indicated $\mathbb{Z}_2$-symmetry lifts to a $\mathbb{Z}_2$-symmetry of $\Lambda(A_{2n-1})$, which has precisely $\binom{2n}{n}$ exact Lagrangian $\mathbb{Z}_2$-fillings.
2. If $\Lambda(\beta)=\Lambda(D_{n+1})$, the indicated $\mathbb{Z}_2$-symmetry lifts to a $\mathbb{Z}_2$-symmetry of $\Lambda(D_{n+1})$, which has precisely $\binom{2n}{n}$ exact Lagrangian $\mathbb{Z}_2$-fillings.
3. If $\Lambda(\beta)=\Lambda(E_6)$, the indicated $\mathbb{Z}_2$-symmetry lifts to a $\mathbb{Z}_2$-symmetry of $\Lambda(E_6)$, which has precisely $105$ exact Lagrangian $\mathbb{Z}_2$-fillings.
4. If $\Lambda(\beta)=\Lambda(D_4)$, the indicated $\mathbb{Z}_3$-symmetry lifts to a $\mathbb{Z}_3$-symmetry of $\Lambda(D_4)$, which has precisely $8$ exact Lagrangian $\mathbb{Z}_3$-fillings.

This extends the ADE perspective to the folded types $B_n$, $C_n$, $F_4$, and $G_2$, where the additional symmetry is part of the classification. The statement is presented as plausible rather than established, so the status of these enumerations remains open.