Milman-moduli characterization of the zeta-minus and zeta-plus moduli

Let $X$ be an arbitrary Banach space. For positive $\varepsilon$, let $\zeta^{-}_{X}$ and $\zeta^{+}_{X}$ be the relevant moduli, and let Milman's moduli be

$$\beta^{-}_{X}(\varepsilon)=\inf_{x,y\in\partial\mathfrak{B}_{1}(o)}\left\{\max\{\lVert x+\varepsilon y\rVert,\lVert x-\varepsilon y\rVert\}-1\right\},$$ $$\beta^{+}_{X}(\varepsilon)=\sup_{x,y\in\partial\mathfrak{B}_{1}(o)}\left\{\min\{\lVert x+\varepsilon y\rVert,\lVert x-\varepsilon y\rVert\}-1\right\}.$$

Milman-moduli conjecture. For positive $\varepsilon$,

$$\zeta^{-}_{X}(\varepsilon)-1=\beta^{-}_{X}(\varepsilon)\quad\text{and}\quad \zeta^{+}_{X}(\varepsilon)-1=\beta^{+}_{X}(\varepsilon).$$

The claim follows the authors' observation that in the definitions of Milman's moduli it should suffice to take only $y\perp x$ in the sense intended by the source. No proof or resolution is supplied.