Singular Gauduchon Conjecture

In 1984 Gauduchon conjectured that one can find Gauduchon metrics with prescribed Ricci curvature on all compact complex manifolds. This conjecture was settled by Sz\'ekelyhidi-Tosatti-Weinkove (TW17, TW19, STW17) by the study of the Monge-Amp\`ere equation for $(n-1)$-plurisubharmonic functions with a gradient term. In this paper we study a singular version of this conjecture. We obtain a $C^{0}$-estimate for this problem, without gradient terms, in smoothable hermitian variaties by adapting a recent technique of Guedj-Lu. We also prove the smoothness of solutions on holomorphic K\"ahler families, generalizing TW17.