Finiteness conjecture for nontrivial Floer cohomology in a Weinstein family

Let $L\subset X$ be a Lagrangian submanifold. In a Weinstein neighborhood, identify a neighborhood of the zero section in $T^*L$, and for $u$ in a small neighborhood of $0$ in $H^1(L;\mathbb R)$ let $L(u)$ be the graph of a closed one-form representing $u$.

Finiteness conjecture for Weinstein deformations. Suppose $QH_{\frak b}(X;\Lambda)$ is semi-simple. Then there exist only finitely many $u$ for which there exists $b\in\mathcal M_{\rm weak}(L(u);\frak b)$ such that

$$HF((L(u),b),(L(u),b);\Lambda)\ne0.$$

This predicts that semisimplicity restricts a fixed Lagrangian's nearby Weinstein deformations with nonzero self-Floer cohomology to a finite set. The supplied text gives no resolution status.