Toric degenerations of Cox rings

This question paraphrases Problem 5.4 in [1]. It was studied in [2] but the authors left it largely unresolved. What are all the toric degenerations of Cox rings of cubic surfaces?

Bibliography
1. Sturmfels, B.; Xu, Z. Sagbi bases of Cox–Nagata rings. Journal of the European Mathematical Society 012.2 (2010): 429-459. // [http://eudml.org/doc/277625].
2. Guillén M.B., Corey D., Donten-Bury M., Fujita N., Merz G. Khovanskii Bases of Cox–Nagata Rings and Tropical Geometry. Fields Institute Communications, vol 80. Springer, New York, NY, 2017//[https://link.springer.com/chapter/10.1007%2F978-1-4939-7486-3_8].
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