Picard-groupoid refinement conjecture for derived Looijenga line bundles

Let $\operatorname{Bil}\colon \operatorname{Lat}^{\operatorname{op}}\to\operatorname{AbGroup}\subset\mathrm{PicGrpds}$ be the functor of bilinear forms, and let $\mathcal{P}ic^{\mathrm{top}}\colon \operatorname{Lat}^{\operatorname{op}}\to\mathrm{PicGrpds}$ assign the Picard groupoid of derived line bundles. Picard-groupoid refinement conjecture. There is an essentially unique natural transformation $\mathcal{L}$ from $\operatorname{Bil}$ to $\mathcal{P}ic^{\mathrm{top}}$ sending $vw\in\operatorname{Bil}(\mathbb{Z})$ to $\mathcal{O}^{\mathrm{top}}(e)[-2]$. In particular, it gives canonical equivalences $\mathcal{L}_{b+b'}\simeq\mathcal{L}_b\otimes\mathcal{L}_{b'}$ and $\mathcal{L}_0\simeq\mathcal{O}_{\mathcal{E}^{\mathrm{top}}\otimes\Lambda}$, hence canonically $\mathcal{L}_{-b}\simeq\mathcal{L}_b^{\vee}$.