Buchanan–odd-cover conjecture for complete graphs

For a graph $G$, let $b_2(G)$ denote its biclique partition number over the field of two elements, and let $K_n$ denote the complete graph on $n$ vertices. Let $k$ be a positive integer. Buchanan–odd-cover conjecture.

$$b_2(K_{2k+1})=k+1,$$

and

$$b_2(K_{2k})=k+1$$

whenever $k\equiv 2 \pmod{4}$ or $k\equiv 3 \pmod{4}$. This extends the known values for complete graphs in the odd-cover problem; the conjecture remains open in the supplied source.