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Henneke F, Kielak D. The agrarian polytope of two-generator one-relator groups. Journal of the London Mathematical Society. Second Series . 2020;102(2):722-748. ; Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Luck-Tillmann.