On adjoint double shuffle relations and adjoint reduction relations

In this paper, we give a positive answer to Jarossay’s question under the adjoint reduction relations. Namely, we prove that the extended double shuffle relations together with the relation $\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}{\pi }^2$ are equivalent to the adjoint double shuffle relations together with the adjoint reduction relations.