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.