The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp{a}lainen and Balazs recently proved this conjecture for the nearest neighbor case. We extend their resu
lts to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.
