The braid-group action conjecture for link Floer chain complexes

Let $(Y,L,p)$ be a pointed link, let $\rho(\sigma_i)$ denote the automorphism induced by the braid-group generator $\sigma_i$, and let $\Psi_i$ and $\Phi_i$ be the maps defining the basepoint-moving automorphism on $\mathit{CFL}(Y,L,p)$. Write $N(K(\mathcal{A}_l))$ for the relevant homotopy category and $\operatorname{Aut}_{N(K(\mathcal{A}_l))}(\mathit{CFL}(Y,L,p))$ for its automorphism group.

Braid-group action conjecture. The two automorphisms $\rho(\sigma_i)$ and $\operatorname{Id}+\Psi_i\Phi_i$ in $\operatorname{Aut}_{N(K(\mathcal{A}_l))}(\mathit{CFL}(Y,L,p))$ are equal.

This conjecture predicts that the automorphism arising from moving the basepoint agrees with the automorphism associated to the braid-group action. It is motivated in the source by the preceding theorem on the induced action, but no resolution is supplied here.