The sphericity conjecture for dichotomous ordinal complete graphs

Let $\mathrm{pdd}(n)$ denote the minimum dimension required to realize a dichotomous ordinal complete graph on $n$ vertices, and let $\mathrm{sp}(G)$ be the sphericity of a graph $G$, namely the least dimension in which $G$ has a unit ball representation. Write $K_{a,b}$ for the complete bipartite graph with parts of sizes $a$ and $b$. Sphericity conjecture.

$$\mathrm{pdd}(n)=\mathrm{sp}(K_{\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor}).$$

The complete bipartite graph is a dichotomous ordinal graph and therefore gives a lower bound for $\mathrm{pdd}(n)$. The conjecture asserts that the balanced complete bipartite graph is the most difficult such graph to realize, but the source reports no results establishing it.