Among all bi-polyhedral Kähler affine structures of type $(\lambda,\nu)$ we expect to find a unique distinguished representative – the Monge-Ampère structure: in affine coordinates the metric satisfies $\det g_{ij}=c$. Its metric completion to $\Sigma$ is supposed to be the limit of the Ricci-flat metrics on the families of Calabi-Yau hypersurfaces.

Note that the Monge-Ampère constant $c$ is determined from calculating the metric volume of $\Sigma$ as $\sqrt{c}\cdot\operatorname{Vol}\partial\Delta^{\vee}_{\lambda}=\sqrt{c^{-1}}\cdot\operatorname{Vol}\partial{\Delta_{\nu}}$, where $\operatorname{Vol}$ means the affine volume of the corresponding polytopal complex. Also note that the rescaled data $(\epsilon^{-1}\lambda,\epsilon\nu,K_{\alpha}(\epsilon y_{\alpha}),\hat{K}_{\alpha}(\epsilon^{-1}\hat{y}_{\alpha}),\epsilon^{2}g_{ij})$ provides the Monge-Ampère structure in the class $[\epsilon^{-1}\lambda,\epsilon\nu]$ with the same metric on $\Sigma$. Thus the Monge-Ampère bi-PIKAS fit together into a projective family.

We will not address this conjecture any further here, rather we will be happy to start with any bi-polyhedral Kähler affine structure on ${\Sigma\backslash D}$ given by a pair $(\Phi,\hat{\Phi})$.