In 1984 Gauduchon conjectured that one can find Gauduchon metrics with prescribed Ricci curvature on all compact complex manifolds. This conjecture was settled by Székelyhidi-Tosatti-Weinkove in 2019. In this talk I will present a singular version of this conjecture for degenerating families and discuss a few first results towards its solution. I will describe the (n-1) Monge-Ampère equation, a fully non-linear elliptic PDE at the heart of this problem. As time allows I will prove that the potential for a degenerate (n-1) Monge-Ampère equation on a compact Hermitian smoothable variety with log-terminal isolated singularities is uniformly bounded and that on the holomorphic deformation of a Kähler Calabi-Yau manifold we can construct a non-Kähler Calabi-Yau-Gauduchon metrics, inside fixed Gauduchon classes, with uniform bounds independent of the complex structure.