Lagrangian-cylinder generation conjecture for the wrapped Fukaya category

Let $V$ be the Liouville sector and let $\mathcal L_\gamma$ and $\mathbb{T}_\gamma$ be the Lagrangian cylinder and the torus obtained from it by wrapping and resolving the resulting intersections, as described above. Lagrangian-cylinder generation conjecture. As an object of $\mathcal{W}(V)$, $\mathbb{T}_\gamma$ is (split-)generated by $\mathcal L_\gamma$. This is motivated by the wrapping construction of $\mathbb{T}_\gamma$ from $\mathcal L_\gamma$ and a slight push-off; the source does not establish the claimed split-generation.