Length of the longest common subsequence between overlapping words

Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected length of an LCS is approximately $\max(\ell, \mathbb{E}[L_n])$, where $L_n$ is the length of an LCS between two independent random sequences. We also obtain tail bounds on this quantity.