Generalized Hadwiger conjecture for covering numbers

Let $B$ be a $d$-dimensional convex body, and let $g_\alpha(B)$ be the infimum of the sums of the $\alpha$-th powers of coefficients of finitely many translative homothets of $B$ with coefficients in $(0,1)$ that cover $B$.

Generalized Hadwiger conjecture. For every $d$-dimensional convex body $B$ and every integer $\alpha\in[0,d]$,

$$g_\alpha(B)\leq 2^{d-\alpha}.$$

For the unit cube, the paper establishes equality for the covering number while showing that the analogous illumination bound fails for almost all values of $\alpha$ and $d$. The generalized covering conjecture remains open in general.