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Bukh and Zhou conjectured that the expectation of the length of the longestcommon subsequence of two i.i.d random permutations of size $n$ is greater than$\sqrt{n}$. We prove in this paper that there exists a universal constant $n_1$such that their conjecture is satisfied for any pair of i.i.d randompermutations of size greater than $n_1$ with distribution invariant underconjugation.  We prove also that asymptotically, this expectation is at least of order$2\sqrt{n}$ which is the asymptotic behaviour of the uniform setting. Moregenerally, in the case where the laws of the two permutations are notnecessarily the same, we gibe a lower bound for the expectation. In particular,we prove that if one of the permutations is invariant under conjugation andwith a good control of the expectation of the number of its cycles, thelimiting fluctuations of the length of the longest common subsequence are ofTracy-Widom type. This result holds independently of the law of the secondpermutation.

On the longest common subsequence of independent random permutations invariant under conjugation