Symplectic four-manifolds and hyperbolic $\pi$-orbifolds

We construct closed symplectic 4-manifolds admitting the structure of hyperbolic $\pi$-orbifold, addressing LeBrun’s conjecture. Our simplest example has Euler characteristic $304$, is tessellated by $64$ copies of the right-angled hyperbolic $120$-cell and admits a four-fold branched covering over a product of two surfaces of genus $4$.