The odd-clique b2 conjecture for odd covers

Let $b_2(G)$ denote the minimum dimension of a binary vector space in which the vertices of a graph $G$ can be represented as an odd cover, and let $K_m$ be the complete graph on $m$ vertices. Odd-clique conjecture. For every integer $n\geq 2$,

$$b_2(K_{2n-1})=n.$$

The source already proves this equality when $2n-1\equiv\pm1\pmod 8$ and conjectures it also for the remaining odd residue classes $2n-1\equiv3,5\pmod 8$. Thus the conjectural content concerns those cases.