Coincidence of asymptotic coefficients with the coefficients of the mean $L_c$

Let $M$ be a symmetric homogeneous mean that has an asymptotic power series expansion, and suppose that it fulfills the requirements of the open question from Farhi. Let $L_c$ denote the mean parameterized by $c\in[-1,+\infty)$. Coefficient coincidence conjecture. The coefficients in the asymptotic power series expansion of $M$ are necessarily the same as those of $L_c$ for some $c\in[-1,+\infty)$. The conjecture proposes uniqueness of the asymptotic coefficients within the class of symmetric homogeneous means satisfying Farhi's requirements; its resolution is not established in the supplied text.