Choi et al.'s optimal-code characterization of the LCP lower bound

Let $n>k\geq 1$. A binary even-like code is a binary linear code whose codewords have even Hamming weight, and ${\bf1}$ denotes the all-one vector. Let $d_L(n,k)$ be the largest minimum distance of a binary $[n,k]$ code, and let $d_{LCP}(n,k)$ be the corresponding largest security parameter for linear complementary pairs of codes.

Choi et al.'s conjecture. There exists a unique binary optimal $[n,k,d_L(n,k)]$ code $C$ which is even-like and contains ${\bf1}$ if and only if

$$d_{LCP}(n,k)=d_L(n,k)-1.$$

Carlet et al. established the bounds $d_L(n,k)-1\leq d_{LCP}(n,k)\leq d_L(n,k)$, while the cited work gives a sufficient condition for attaining the lower bound. The conjecture asserts that the stated uniqueness and structural condition is also necessary.