Bounded-dimensional orthostochasticity conjecture

Let ${\mathcal B}_n$ denote the set of bistochastic $n\times n$ matrices, and let $d_{\min}(n)$ be the smallest value of $d$ such that every matrix in ${\mathcal B}_n$ is $d$-orthostochastic. Bounded-dimensional orthostochasticity conjecture. There exists $n_0$ such that, for every $n\ge n_0$,

$$2\le d_{\min}(n)\le 3.$$

This conjecture asserts that the number of real dimensions needed to represent all bistochastic matrices by orthostochastic constructions remains bounded as the matrix size grows. The surrounding discussion presents it as plausible based on the preceding results and dimensional considerations; its resolution is not given in the source.