  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 for
the integrated $\varphi$-scalar curvature functional restricted to metrics with
unit volume and constant $\varphi$-scalar curvature. We prove both a rigidity
statement for solutions to $(\varphi-\mathrm{CPE})$ in a conformal class, and a
gap theorem characterizing the round sphere among manifolds supporting
$(\varphi-\mathrm{CPE})$ with $\varphi$ a harmonic map.
