Self-complementary vertex-transitive graph conjecture

Let $G=(V,E)$ be a self-complementary, vertex-transitive graph with $|V|=n$. The self-complementary vertex-transitive conjecture. Its orthogonal representation dimension is

$$\rho ^\perp(G)=\lceil n/2\rceil.$$

The preceding proposition establishes the lower bound for self-complementary graphs, while the source reports no proof of equality and notes that the hypotheses cannot generally be weakened.