Soltan's covering conjecture for convex bodies

Let $B\subset \mathbb{R}^d$ be a convex body. Define

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

Soltan's conjecture. For every natural number $d$,

$$g_1(d)\geq d.$$

The conjecture is known for $d=2$, asymptotically as $d\to\infty$, and for the $d$-dimensional Euclidean ball, but remains open in general.