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Let $A$ be a noncommutative Artin–Schelter regular algebra of dimension $2$ with the Nakayama automorphism $\mu _A$ and $U$ a PBW deformation of $A$ with the Nakayama automorphism $\mu _U$. We prove that any graded Ore extension $A[z;\mu _A,\delta ]$ and any filtered Ore extension $U[z;\mu _U,\tilde{\delta }]$ are $3$-Calabi–Yau.