Lagrangianity conjecture for the automorphism group of an A-infinity category

Let ${\cal C}$ be an $A_{\infty}$-category such that $HH^1({\cal C})$ is abelian and the connected automorphism group $G_{\cal C}=Aut^0({\cal C})$ is a finite-dimensional commutative Lie group over ${\bf C}$. Let $L_{\cal C}=H_1(G_{\cal C},{\bf Z})$, equipped with the symmetric bilinear form induced by the first Chern class of the ${\bf C}^{\ast}$-torsor $P_{\cal C}$, and let $\Lambda_{\cal C}$ be the kernel of the linear map $p:L_{\cal C}\otimes {\bf C}\to g_{\cal C}$. Lagrangianity conjecture. The symmetric bilinear form $(x,y)$ is non-degenerate, and the subspace $\Lambda_{\cal C}$ is maximal isotropic, that is, Lagrangian, with respect to it. This conjectural structure connects the automorphism group and Hochschild cohomology of an $A_{\infty}$-category with a quantum-torus-type lattice; the supplied text gives no resolution of the claim.