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This is largely an exposition article that expands the author's talk at theXiamen International Conference on Geometric Analysis in June 2021. We firstsurvey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD)conjecture for $g$-weighted K\"{a}hler-Ricci solitons ($g$-solitons). We thenreview recent works of Apostolov-Canderbank-Jubert-Lahdili which establish aconnection between a particular $g$-soliton equation with Ricci-flat K\"{a}hlercone metrics (or equivalently Sasaki-Einstein metrics). The main interest inthis connection is the transformation of a possibly irregular Sasaki-Einsteinmetric to a particular $g$-soliton equation on any quasi-regular quotient. Wewill revisit this transformation by understanding how the correspondingtransversal complex Monge-Amp\`{e}re equations are transformed under thedeformation of Reeb vector fields. Finally we explain how thisPDE/pluripotential point of view allows one to prove the YTD conjecture forgeneral Fano cones by combining the YTD conjecture for $g$-solitons on log Fanopairs and the above transformation.

Notes on weighted K\"{a}hler-Ricci solitons and application to Ricci-flat K\"{a}hler cone metrics