  In a seminal paper, Kannan and Lov\'asz (1988) considered a quantity
$\mu_{KL}(\Lambda,K)$ which denotes the best volume-based lower bound on the
covering radius $\mu(\Lambda,K)$ of a convex body $K$ with respect to a lattice
$\Lambda$. Kannan and Lov\'asz proved that $\mu(\Lambda,K) \leq n \cdot
\mu_{KL}(\Lambda,K)$ and the Subspace Flatness Conjecture by Dadush (2012)
claims a $O(\log(2n))$ factor suffices, which would match the lower bound from
the work of Kannan and Lov\'asz.
  We settle this conjecture up to a constant in the exponent by proving that
$\mu(\Lambda,K) \leq O(\log^{3}(2n)) \cdot \mu_{KL} (\Lambda,K)$. Our proof is
based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz
(2017). Following the work of Dadush (2012, 2019), we obtain a
$(\log(2n))^{O(n)}$-time randomized algorithm to solve integer programs in $n$
variables. Another implication of our main result is a near-optimal flatness
constant of $O(n \log^{3}(2n))$.
