Core--cocore Koszul duality conjecture

Let $X$ be a Weinstein manifold with contact boundary $Y$, let $L=\bigcup_{i\in I}L_i$ be its relative skeleton decomposed into smooth Lagrangians, and let $C_i$ be a cocore or linking disk intersecting $L$ at a point of $L_i$. Write $\operatorname{End}_{\operatorname{Fuk}}(\bigoplus_iL_i)$ for the endomorphism algebra in the infinitesimally wrapped Fukaya category and $\operatorname{End}_{\mathcal W}(\bigoplus_iC_i)$ for that in the partially wrapped Fukaya category.

Core--cocore Koszul duality conjecture.

$$\operatorname{End}_{\operatorname{Fuk}}\left(\bigoplus_iL_i\right)\quad\text{and}\quad \operatorname{End}_{\mathcal W}\left(\bigoplus_iC_i\right)$$

are Koszul dual, expressing a duality between the core (skeleton) and the cocore, or linking disk.

The source explicitly calls this a false conjecture, so the statement is refuted as written. The surrounding discussion presents it as a naive version of the core--cocore philosophy rather than a valid general theorem.