The curved filtered $A_{\infty}$-algebra conjecture for Lagrangian cohomology

Suppose $(X,\omega)$ and $\mathcal D$ are as in the beginning of the introduction, and let $L\subset X\setminus\mathcal D$ be a relatively spin compact Lagrangian submanifold. Let $\Lambda^{{\mathbb Q}}_0$ be the Novikov ring with rational coefficients. Curved filtered $A_{\infty}$-algebra conjecture. There exists a curved filtered $A_{\infty}$-algebra

$$\left(H^*(L;\Lambda^{{\mathbb Q}}_0),\{\frak m_k\mid k=0,1,2,\dots\}\right)$$

on $H^*(L;\Lambda^{{\mathbb Q}}_0)$, independent of choices such as almost complex structures up to homotopy equivalence. If $L$ is monotone, then after setting $T=1$,

$$\frak m_0(1)=\frak{PO}_L(1)1\in H^0(L;\Lambda^{{\mathbb Q}}_0),$$

where $\frak{PO}_L$ is as in Definition and $1\in H^0(L;\Lambda^{{\mathbb Q}}_0)$ is the unit. The conjecture concerns the extension of the monotone Lagrangian Floer construction to curved filtered $A_{\infty}$-structures; the source indicates that it should follow from suitable Kuranishi structures and polygon moduli spaces, but does not establish it here.