Generation conjecture for the HDHF wrapped Fukaya category of cotangent bundles

Let $Q$ be a closed oriented manifold with vanishing relative second Stiefel–Whitney class, and let $\mathcal{W}_\kappa(T^*Q)$ denote its HDHF wrapped Fukaya category. A $\kappa$-tuple of cotangent fibers is a collection of $\kappa$ pairwise disjoint cotangent fibers in $T^*Q$.

Generation conjecture. The category $\mathcal{W}_\kappa(T^*Q)$ is split-generated by any $\kappa$-tuple of disjoint cotangent fibers.

This conjecture proposes the multiple-particle analogue of the generation criterion for the wrapped Fukaya category of a cotangent bundle. The surrounding discussion suggests that algebraic structures on HFSH may help establish it, but no resolution is given here.