Natural equivalence of Dehn and spherical twist of Fukaya category

Question

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$.

A Lagrangian sphere $L\subset M$, equipped with a brane structure, defines an object of $\text{Fuk}(M)$. There are two twist functors associated to $L$. The first is the autoequivalence of the Fukaya category associated to the Dehn twist on $L$. The second is the spherical twist functor of the spherical object $L\in \text{Fuk}(M)$. It is shown by Seidel, see corollary III.17.17, that pointwise both twist functors are equivalent.

Question: Has it been shown in general that there is an equivalence of functors between both twist functors? Have counterexamples been found?

What I know so far: In remark III.17.22, Seidel writes that ongoing work of Wehrheim-Woodward on Lagrangian correspondences is supposed to show that this equivalence is natural. I don't know if this indeed follows from their papers, at least it is not explicitly mentioned. Using very different ideas, the equivalence of functors is, as far as I understand, established in some examples of complex dimension 1 by S. Opper in arXiv:1904.04859.

Answer 1

I believe this is solved (probably the way Seidel meant in his remark 17.22) in this paper: https://arxiv.org/pdf/1509.08028.

Given Lagrangian sphere $L$ in symplectic manifold $(M,\omega)$, they proved the Lagrangian surgery of $L\times L$ and the diagonal $\Delta$ along their (clean) intersection is Hamiltonian isotopic to the graph of $\tau_L^{-1}$ in $(M\times M, \omega\oplus -\omega)$. This shows that there is an exact triangle in $Fuk(M\times M, \omega\oplus -\omega)$:

$L\times L \rightarrow \Delta \rightarrow Gr(\tau_L^{-1})$ Using a trick they proved that the first map represents $\mu_2$. So the exact triangle above, when taking $HF(A\times B,-)$ is equivalent to Seidel's exact sequence on Dehn twists.

Now https://arxiv.org/pdf/1601.04919 gives an $A_\infty$ functor $\Phi$ from $Fuk(M\times M, \omega\oplus -\omega)$ to $Fun(Fuk^\#(M),Fuk^\#(M))$ where $Fuk^\#(M)$ is extended Fukaya category which we can take to be $Fuk^(M)$ because Lagrangian composition with $L\times L$ and graphs are very simple. $\Phi$ maps the exact triangle above to the triangle of functors:

$hom(-,L)\otimes L \rightarrow id \rightarrow \tau_L$

which implies Dehn twist is isomorphic to the algebraic twist