Tropical basis for the universal Cox ideal

This question paraphrases Conjecture 5.3 in [1]. Can we identify a tropical basis for the universal Cox ideal of cubic surfaces?

Bibliography
1. Ren, Q.; Shaw, K.; Sturmfels, B. Tropicalization of del Pezzo surfaces. Advances in Mathematics, Volume 300, 156-189, 2016// [https://www.sciencedirect.com/science/article/pii/S0001870816001213].
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