Itô formula for planarly branched rough paths

The Itô formula, originated by K. Itô, is focus on the stochastic calculus, where many stochastic processes can be placed under the framework of rough paths. In rough path theory, Itô formulas have been proved for rough paths with roughness $\frac{1}{3}<\alpha \le \frac{1}{2}$ and branched rough paths with roughness $0<\alpha \le 1$. Planarly branched rough paths contain more random processes than rough paths and branched rough paths. In the present paper, we prove the Itô formula for planarly branched rough paths with roughness $\frac{1}{4}<\alpha \le \frac{1}{2}$.