Day–Nordlander conjecture for the moduli gamma-minus and gamma-plus

Let $X$ be an arbitrary Banach space. The moduli $\gamma^{-}_{X}$ and $\gamma^{+}_{X}$ are compared with their Hilbert-space values, where $H$ denotes a Hilbert space:

$$\gamma^{-}_{X}(\varepsilon) \leqslant \gamma^{-}_{H}(\varepsilon)=\varepsilon^{2}=\gamma^{+}_{H}(\varepsilon) \leqslant \gamma^{+}_{X}(\varepsilon).$$

Day–Nordlander conjecture. The displayed inequalities hold.

An analogue is known in the infinite-dimensional case using Dvoretzky's theorem, but the finite-dimensional case was open in the source.