An inequality between Jordan–von Neumann constant and James constant

Let XX be a non-trivial Banach space. L. Maligranda conjectured CNJ(X)≤1+J(X)2/4CNJ(X)≤1+J(X)2/4 for James constant J(X)J(X) and von Neumann–Jordan constant CNJ(X)CNJ(X) for XX. Recently, J. Alonso et al. gave a proof of it and conjectured that CNJ(X)≤J(X)CNJ(X)≤J(X) is also valid. In this paper, we show that this conjecture is true.