One application of Calabi-Ansatz is choosing inital metric to run the Kahler–Ricci flow. So if you want to use minimal model program and apply Kahler–Ricci flow to find canonical metric study of such Calabi-Ansatz would be very important. In fact study of Calabi-Ansatz gives an effective way to find inital metric and its connection with semi-flat metric. The question is still open.

See the paper of 


Jian Song, Yuan Yuan, [Metric Flips with Calabi Ansatz, Geometric and Functional Analysis](https://doi.org/10.1007/s00039-012-0151-1),
Geometric and Functional Analysis, February 2012, Volume 22, Issue 1, pp 240–265.


See proposition 3.2 of [Futaki - Momentum construction on Ricci-flat Kähler cones](https://doi.org/10.2748/tmj/1303219934) for study of Sasaki–Ricci flow and finding suitable inital metric via Calabi-Ansatz.


  [1]: https://link.springer.com/article/10.1007/s00039-012-0151-1
  [2]: https://projecteuclid.org/download/pdf_1/euclid.tmj/1303219934