Codenotti–Santos–Schymura conjecture on covering minima of weighted terminal simplices

Let $\omega=(\omega_0,\dots,\omega_d)\in\mathbb{R}^{d+1}_{>0}$ satisfy $\omega_0\leqslant\dots\leqslant\omega_d$, and define the weighted terminal simplex

$$S(\omega):=\operatorname{conv}(-\omega_0\mathbb{1}_d,\omega_1e_1,\dots,\omega_de_d)\subseteq\mathbb{R}^d.$$

For $i\in[d]$, Codenotti–Santos–Schymura conjecture. The $i$-th covering minimum of $S(\omega)$ is attained by projection to the first $i$ coordinates, namely

$$\mu_i(S(\omega))=\frac{\sum\limits_{0\leqslant j<k\leqslant i}\frac{1}{\omega_j\omega_k}}{\sum\limits_{j=0}^i\frac{1}{\omega_j}}.$$

The source says that the covering radius of every weighted terminal simplex is known and that this conjecture gives the other covering minima, with the case $i=1$ proved; the remaining cases are open in the supplied text.