Merino–Schymura conjecture on covering minima of terminal simplices

Let $T_d=\operatorname{conv}(-\mathbb{1}_d,e_1,\dots,e_d)$ be the terminal simplex in dimension $d$, and let $i\in[d]$. Merino–Schymura conjecture. For every $d\in\mathbb{N}$ and every $i\in[d]$,

$$\mu_i(T_d)=\frac{i}{2}.$$

The conjecture concerns all covering minima of the terminal simplex; the supplied text does not state a resolution, so the general assertion remains open.