The well-studied vector balancing constant $β(U, V)$ of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, $α(U,V)$. In [BS97], Banaszczyk and Szarek proved that $α(B_2^n, V)\leq c$ when $V$ has Gaussian measure at least $\frac{1}{2}$, and conjectured that, for centrally symmetric $V$, $β(B_2^n, V)$ is always bounded by a function of the Gaussian measure of $V$, independent of $n$. We resolve this conjecture in the affirmative. Moreover, we show that the analogous result holds for $α(B_2^n, V)$ even without the central symmetry assumption.