  This is largely an exposition article that expands the author's talk at the
Xiamen International Conference on Geometric Analysis in June 2021. We first
survey 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 then
review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a
connection between a particular $g$-soliton equation with Ricci-flat K\"{a}hler
cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in
this connection is the transformation of a possibly irregular Sasaki-Einstein
metric to a particular $g$-soliton equation on any quasi-regular quotient. We
will revisit this transformation by understanding how the corresponding
transversal complex Monge-Amp\`{e}re equations are transformed under the
deformation of Reeb vector fields. Finally we explain how this
PDE/pluripotential point of view allows one to prove the YTD conjecture for
general Fano cones by combining the YTD conjecture for $g$-solitons on log Fano
pairs and the above transformation.
