Lexicode conjecture for clique covering numbers of Johnson graphs

Let $J(2k,k-1)$ be the Johnson graph whose vertices are the $(k-1)$-subsets of a $2k$-element set, and let $JK(2k,k-1)$ be the graph used in the construction above. Order the vertices of $J(2k,k-1)$ lexicographically and apply the greedy algorithm to obtain a lexicode

in $J(2k,k-1)$. Let $C_k$ denote the corresponding benchmark cardinality. **Lexicode conjecture.** Assume that $k>1$ is a power of $2$. The lexicode

has cardinality

$$||=\tfrac{1}{2}C_k,$$

and its elements are pairwise overlapping. In particular, the code defines an independent set of cardinality $C_k$ in $JK(2k,k-1)$, so that

$$\theta(J(2k,k))=C_k.$$

The construction is known computationally for $k=2,4,8,16$. The conjecture would establish the stated clique-covering value for every $k>1$ that is a power of $2$.