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In this paper, we study an extension of the CPE conjecture to manifolds $M$which support a structure relating curvature to the geometry of a smooth map$\varphi : M \to N$. The resulting system, denoted by $(\varphi-\mathrm{CPE})$,is natural from the variational viewpoint and describes stationary points forthe integrated $\varphi$-scalar curvature functional restricted to metrics withunit volume and constant $\varphi$-scalar curvature. We prove both a rigiditystatement for solutions to $(\varphi-\mathrm{CPE})$ in a conformal class, and agap theorem characterizing the round sphere among manifolds supporting$(\varphi-\mathrm{CPE})$ with $\varphi$ a harmonic map.

Einstein-type structures, Besse's conjecture and a uniqueness result for a $\varphi$-CPE metric in its conformal class