Generalized Soltan conjecture for translative homothetic coverings

Let $B\subset\mathbb{R}^d$ be a 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 $B$. Set

$$g_\alpha(d)=\inf\left\{g_\alpha(B): B\subset\mathbb{R}^d,\ B\text{ is a convex body}\right\}.$$

Generalized Soltan conjecture. For all natural $d$ and all $\alpha$ such that $0\leq\alpha\leq d+1$,

$$g_\alpha(d)=d+1-\left\lceil\alpha\right\rceil.$$

The formula was proposed as a generalization of Soltan's conjecture and is tight for Euclidean balls if true; its general validity remains open.