Generalized Fukaya conjecture for polarized Kähler manifolds

Let $X$ be a closed Kähler manifold, and let $\Sigma\subset X$ be a complex hypersurface such that $X\setminus\Sigma$ is a Weinstein manifold admitting a cyclic dilation; then $(X,\Sigma)$ is a polarization with finite first Gutt-Hutchings capacity. Let $L\subset X$ be a closed Lagrangian submanifold that is a $K(\pi,1)$ space and $\mathit{Spin}$. Generalized Fukaya conjecture. There exists $\bar a\in\pi_2(X,L)$ such that $L$ bounds a holomorphic disc in class $\bar a$ with Maslov index $2$; moreover, $a=\partial\bar a\in\pi_1(L)$ is nonzero and its centralizer $Z_a\subset\pi_1(L)$ has finite index. This extends the stated theorem for $\mathbb{CP}^n$ to polarized Kähler manifolds and is presented as a natural expectation; no proof or resolution is supplied.