  Hadwiger's conjecture in convex geometry, formulated in 1957, states that
every convex body in $\mathbb{R}^n$ can be covered by $2^n$ translations of its
interior. Despite significant efforts, the best known bound related to this
problem was $\mathcal{O}(4^n \sqrt{n} \log n)$ for more than sixty years. In
2021, Huang, Slomka, Tkocz, and Vritsiou made a major breakthrough by improving
the estimate by a factor of $\exp\left(\Omega(\sqrt{n})\right)$. Further, for
$\psi_2$ bodies they proved that at most $\exp(-\Omega(n))\cdot4^n$
translations of its interior are needed to cover it.
  Through a probabilistic approach we show that the bound
$\exp(-\Omega(n))\cdot4^n$ can be obtained for convex bodies with sufficiently
many well-behaved sub-gaussian marginals. Using a small diameter approximation,
we present how the currently best known bound for the general case, due to
Campos, Van Hintum, Morris, and Tiba can also be deduced from our results.
