Duality pairing conjecture for better-behaved GKZ systems

Let $C$ be a rational polyhedral cone in a lattice $N$, let $C^\circ$ denote the associated dual data, and let $bbGKZ(C,0)$ and $bbGKZ(C^\circ,0)$ be the corresponding better-behaved GKZ systems. Let $(\Phi_c)$ and $(\Psi_d)$ be solutions of these systems, indexed by $c\in C$ and $d\in C^\circ$, respectively. Duality pairing conjecture. There exists a collection of polynomials $p_{c,d}(x_1,\ldots,x_n)$, indexed by $c\in C$ and $d\in C^\circ$, such that all but finitely many are zero; for every pair of solutions $(\Phi_c)$ and $(\Psi_d)$, the sum

$$\sum_{c,d}p_{c,d}\Phi_c\Psi_d$$

is constant as a function of $(x_1,\ldots,x_n)$; the resulting pairing is non-degenerate; and for every projective simplicial subdivision $\Sigma$, it is the inverse of the Euler-characteristics pairing between $K_0(\mathbb{P}_{\Sigma})$ and $K^c_0(\mathbb{P}_{\Sigma})$ under the $\Gamma$ and $\Gamma^\circ$ series. This conjecture proposes a specific duality mechanism between the two better-behaved GKZ systems, motivated by prior expectations that these systems are dual; its validity was described as plausible but unsettled.