An isoperimetric inequality for word overlap

Let $A$ and $B$ be sets of words of length $n$ over some finite alphabet. Suppose that no suffix of a word in $A$ coincides with a prefix of a word in $B$. Then we show that the product of densities of $A$ and $B$ is upper bounded by $(1+o(1))\mathrm{/}(en)$. This bound is asymptotically sharp.