Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index 0. Consider a collection of Lagrangian submanifolds $L_1,\cdot\cdot\cdot,L_k\subset X$, and assume that $L_i\cong\mathbb{R}^n$. Since $H^1(L_i)=0$, $HF^\ast(L_i,L_j)$ is well-defined and $\mathbb{Z}$-graded for arbitrary $i$ and $j$. Denote by $\mathscr{F}^\bigstar(X)$ the $A_\infty$ category whoses objects consists of all the closed exact Lagrangian submanifolds with vanishing Maslov index together with the contractible Lagrangians $\{L_1,\cdot\cdot\cdot,L_k\}$ introduced above. My question is when $\{L_1,\cdot\cdot\cdot,L_k\}$ split-generates $D^\pi\mathscr{F}^\bigstar(X)$?
A well-known special case of my question is the work of Seidel, which concerns the question above in the case when $\{L_1,\cdot\cdot\cdot,L_k\}$ is a certain collection of Lefschetz thimbles. In the absence of a Lefschetz fibration, is there any concise criterion on the collection $\{L_1,\cdot\cdot\cdot,L_k\}$ to get the required split-generation result?
