This is answered in this paper, based on a construction of Gromov-Thurston (as commented by @YangMills):

<cite authors="Fine, Joel; Premoselli, Bruno">_Fine, Joel; Premoselli, Bruno_, [**Examples of compact Einstein four-manifolds with negative curvature**](http://dx.doi.org/10.1090/jams/944), J. Am. Math. Soc. 33, No. 4, 991-1038 (2020). [MR4155218](https://mathscinet.ams.org/mathscinet-getitem?mr=4155218) [ZBL1467.53055](https://zbmath.org/?q=an:1467.53055).</cite>