The even-like extremal binary LCD code conjecture

Let $k$ be an even positive integer and let $n>k$ be an integer. Write $d_2^{E}(n,k)$ for the relevant extremal minimum distance of binary LCD codes, and call a binary LCD code even-like when every codeword has even Hamming weight. The even-like extremal-code conjecture. If $d_2^{E}(n,k)$ is even and

$$d_2^{E}(n-1,k)=d_2^{E}(n,k)-1,$$

then every binary LCD $[n,k,d_2^{E}(n,k)]$ code is even-like. The paper states that this conjecture is invalid and provides two counterexamples.