This is answered in this paper, based on a construction of Gromov-Thurston (as commented by @YangMills):
Fine, Joel; Premoselli, Bruno, Examples of compact Einstein four-manifolds with negative curvature, J. Am. Math. Soc. 33, No. 4, 991-1038 (2020). MR4155218 ZBL1467.53055.