The cube-root fluctuation conjecture for the LCS concatenation gain

Let $V$ and $W$ be independent random words of length $2n$, written as concatenations $V=V_1V_2$ and $W=W_1W_2$, where each component has length $n$. Define

$$\Delta(V,W)=\operatorname{LCS}(V,W)-\operatorname{LCS}(V_1,W_1)-\operatorname{LCS}(V_2,W_2),$$

and write $\Delta(2n)=\Delta(V,W)$. Cube-root fluctuation conjecture. There are constants $c_1,c_2$ such that

$$\operatorname{\mathbb{E}}\Delta(n)\sim c_1n^{1/3}$$

and

$$\sqrt{\operatorname{Var}\Delta(n)}\sim c_2n^{1/3}.$$

The conjecture is suggested by the paper’s computational experiments on the excess LCS obtained by allowing matches to cross the two concatenated blocks. The cited prior estimate gives only the upper-order control $\operatorname{\mathbb{E}}\Delta(V,W)=O(\sqrt{n\log n})$, so the cube-root asymptotics remain open.