Longest common subsequences between words of very unequal length

We consider the expected length of the longest common subsequence between two random words of lengths $n$ and $(1-varepsilon)kn$ over $k$-symbol alphabet. It is well-known that this quantity is asymptotic to $gamma_{k,varepsilon} n$ for some constant $gamma_{k,varepsilon}$. We show that $gamma_{k,varepsilon}$ is of the order $1-cvarepsilon^2$ uniformly in $k$ and $varepsilon$. In addition, for large $k$, we give evidence that $gamma_{k,varepsilon}$ approaches $1-tfrac{1}{4}varepsilon^2$, and prove a matching lower bound.