Necessity of the conditions in the sufficiency theorem for vector-valued Runge–Kutta methods

Let a vector-valued explicit or implicit Runge–Kutta method have order $p$, and let the conditions in the sufficiency theorem referred to in the source be imposed on such a method. Necessity conjecture. The set of conditions in the sufficiency theorem is not only sufficient but also necessary for a vector-valued explicit or implicit Runge–Kutta method to achieve order $p$. The conjecture concerns whether the stated simplifying conditions characterize order $p$ for vector-valued Runge–Kutta methods; the source presents this as an unresolved question about the necessity of its sufficiency-theorem conditions.