Pandichotomic dimension of complete graphs and balanced complete bipartite sphericity

Prove or refute the conjecture that the smallest Euclidean dimension in which every dichotomous ordinal graph on $n$ vertices is realizable equals the sphericity of the balanced complete bipartite graph $K_{\lceil n/2\rceil,\lfloor n/2\rfloor}$, namely establish whether $\mathrm{pdd}(n)=\mathrm{sp}(K_{\lceil n/2\rceil,\lfloor n/2\rfloor})$.

Background

In the additional discussion of complete graphs, $\mathrm{pdd}(n)$ is defined as the smallest Euclidean dimension in which the complete graph $K_n$, and hence every graph on $n$ vertices, is pandichotomous. The sphericity $\mathrm{sp}(G)$ is the least dimension in which a graph is representable as a unit-ball graph.

The text observes that the sphericity of balanced complete bipartite graphs gives lower bounds for $\mathrm{pdd}(n)$ and conjectures that these graphs are the extremal obstructions determining the exact value. Establishing the conjecture would connect the worst-case dichotomous realization dimension for $n$ vertices with a specific complete bipartite sphericity.