In 1987 Gromov and Thurston developed the first Riemannian manifolds that are not homotopy equivalent to a hyperbolic manifold but admit a Riemannian metric that is ϵ-pinched for any given ϵ>0. The manifolds that they construct are branched covers of hyperbolic manifolds, and to construct the metric they perform a sort of “geometric surgery” about the ramification locus. In 2022 Stover and Toledo proved the existence of similar branched cover manifolds built out of complex hyperbolic manifolds, and via a result of Zheng these manifolds admit a negatively curved Kahler metric. In this talk we will discuss how to construct a (not Kahler) Riemanain metric on these Stover-Toledo manifolds which is ϵ-close to being 1/4-pinched for any prescribed ϵ>0. These provide the first known examples of Kahler manifolds that are not homotopy equivalent to a complex hyperbolic manifold but admit a Riemannian metric that is ϵ-close to being 1/4-pinched.