A partial verification of Conjecture 3.6 in the case $M=T^{2}$ is provided by Haug [28]. He defines a version of the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ in which the objects are twisted complexes built out of pairs $(L,E)$ for $L$ a compact, spin, graded, embedded Lagrangian in $T^{2}$, and $E\rightarrow L$ a local system, and proves that $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ is idempotent complete.

Haug remarks [28, §1] that for $T^{2}$, including local systems $E\rightarrow L$ has the effect of making $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ idempotent complete, and that $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ would not be idempotent complete if we took objects to be twisted complexes of Lagrangians $L$ rather than pairs $(L,E)$. This shows that including local systems $E\rightarrow L$ in objects $(L,E,b)$ is necessary for our programme, since otherwise Conjecture 3.6 and hence Conjecture 3.2 would be false even for $M=T^{2}$. We will see in §3.4 how nontrivial local systems are needed for some kinds of surgeries.

Haug’s definition of $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ is not quite the same as ours. He does not include bounding cochains $b$ in his objects $(L,E)$ (the simplicity of dimension 1 permits this). He fixes ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{C}}}$. His local systems $E\rightarrow L$ [28, §3.1.1] are not ${\mathbin{\mathbb{F}}}$-local systems, as in §2.5, but $\Lambda_{\rm nov}$-local systems of arbitrary finite rank, such that (roughly) the eigenvalues of $\mathop{\rm Hol}\nolimits(\nabla_{E})$ lie in ${\mathbin{\mathbb{F}}}^{*}\subset\Lambda_{\rm nov}^{*}$ to leading order.

I expect this should be related to our definition of $D^{b}{\mathbin{\mathscr{F}}}(T^{2})$ as follows. In dimension 1, the combination of a rank one ${\mathbin{\mathbb{F}}}$-local system $E\rightarrow L$ and a bounding cochain $b$ is essentially equivalent to a rank one $\Lambda_{\rm nov}$-local system $E_{\rm nov}\rightarrow L$ satisfying Haug’s condition, where the holonomies satisfy $\mathop{\rm Hol}\nolimits(\nabla_{E_{\rm nov}})[\gamma]=\mathop{\rm Hol}\nolimits(\nabla_{E})[\gamma]\cdot e^{\int_{\gamma}b}$ for $[\gamma]\in\pi_{1}(L)$. Also, I expect that for $T^{2}$, considering rank one local systems $E\rightarrow L$ on immersed Lagrangians has a similar effect to considering higher rank local systems $E\rightarrow L$ on embedded Lagrangians.