Pandichotomic dimension and complete bipartite sphericity

Prove or refute the conjecture that the pandichotomic dimension of the complete graph on $n$ vertices equals the sphericity of the complete bipartite graph with bipartition sizes $\lceil n/2\rceil$ and $\lfloor n/2\rfloor$.

Background

The later material introduces the pandichotomic Euclidean dimension pdd(n)\mathrm{pdd}(n) of the complete graph, and relates lower bounds for this quantity to the sphericity of complete bipartite graphs. It observes that a short complete bipartite subgraph is one of the dichotomous ordinal instances that must be handled when determining pdd(n)\mathrm{pdd}(n).

The proposed equality identifies the balanced complete bipartite graph as determining the worst case for the pandichotomic dimension. It is stated as a formal conjecture and is not proved in the supplied text.

References

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

— Geometric realizations of dichotomous ordinal graphs  (2503.07361 - Angelini et al., 10 Mar 2025) in Section Sphericity of $K_{m,m}$ (material following the main document)