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In this paper we construct an almost negatively $1/4$-pinched Riemannianmetric on a class of compact manifolds recently discovered by Stover and Toledoin [17]. It is known that these manifolds are K\"{a}hler and not locallysymmetric. These are the first known examples of not locally symmetricK\"{a}hler manifolds admitting such a metric and, via the result of Hernandez[9] and Yau and Zheng [18], these manifolds cannot admit a negativelyquarter-pinched Riemannian metric. This metric is also interesting because itis a generalization to the complex hyperbolic setting of the famous pinchedmetric constructed by Gromov and Thurston in [8].

K\"{a}hler manifolds with an almost $1/4$-pinched metric