Conformally K\"ahler, Einstein--Maxwell metrics and boundedness of the modified Mabuchi-functional

We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge K\"ahler class a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$-scalar curvature, then this metric minimizes the $(\xi,a,p)$-Mabuchi functional. Our method of proof extends the approach introduced by Donaldson and developed by Li and Sano--Tipler, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of Koca--T{\o}nnesen-Friedman, we describe the K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admit conformally K\"ahler, Einstein-Maxwell metrics.