Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler--Ricci shrinker satisfying \[ \operatorname{Ric}+\nabla^2 f=\frac12 g. \] We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in\{0,\ldots,m\}$ such that \[ R\equiv k \] and \[ (M^