  The various scalar curvatures on an almost Hermitian manifold are studied, in
particular with respect to conformal variations. We show several integrability
theorems, which state that two of these can only agree in the K\"ahler case.
Our main question is the existence of almost K\"ahler metrics with conformally
constant Chern scalar curvature. This problem is completely solved for ruled
manifolds and in a complementary case where methods from the Chern-Yamabe
problem are adapted to the non-integrable case. Also a moment map
interpretation of the problem is given, leading to a Futaki invariant and the
usual picture from geometric invariant theory.
