Higher-dimensional parallelogram decomposition conjecture for centrally symmetric polyhedral surfaces

Let $\delta_1,\delta_2,\dots,\delta_N$ be $N$ positive numbers that sum up to $2\pi$. Let $\overline{\mathcal{M}_{2N}}(\delta_1,\delta_2,\dots,\delta_N)$ denote the metric completion of the moduli space of centrally symmetric convex polyhedral surfaces with $2N$ vertices and the prescribed cone-deficits, equipped with its antipodal map.

Higher-dimensional parallelogram decomposition conjecture. Every surface in

$$\overline{\mathcal{M}_{2N}}(\delta_1,\delta_2,\dots,\delta_N)$$

can be decomposed into at most

$$2\binom{2N-2}{2}$$

parallelograms, and the decomposition is invariant under the antipodal map.

This conjecture generalizes the proved case represented by Theorem 5, where $N=5$ and the bound is $2\binom{8}{2}=56$. The source presents the extension to higher dimensions as an open direction for future work.