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We construct an integrable hierarchy in the form of Hirota quadraticequations (HQE) that governs the Gromov--Witten (GW) invariants of the Fanoorbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators inour construction are given in terms of the $K$-theory of$\mathbb{P}^1_{a_1,a_2,a_3}$ via Iritani's $\Gamma$-class modification of theChern character map. We also identify our HQEs with an appropriateKac--Wakimoto hierarchy of ADE type. In particular, we obtain a generalizationof the famous Toda conjecture about the GW invariants of $\mathbb{P}^1$ .

Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies