On the Coble quartic and Fourier-Jacobi expansion of theta relations

In the paper "The universal Kummer threefold", Q. Ren, S. Sam, G. Schrader, and B. Sturmfels (arXiv:1208.1229), conjectured equations for the universal Kummer variety in genus 3 case. Though, most of these equations are obtained from the Fourier-Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic was obtained differently. The aim of the current paper is to show that Coble's quartic can be obtained as Fourier-Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in "The universal Kummer threefold".