Linear Euclidean realizability conjecture for median systems

Let a median system be a set system whose 1-inclusion graph is a median graph, and let $d$ denote its dimension. Median-system realization conjecture. Every median system of dimension $d$ admits a realization in

$$\mathbb{R}^{O(d)}.$$

Trees provide a two-dimensional realization, and the surrounding discussion relates this claim to the equality of VC-dimension and topological dimension for ample median systems. The asserted linear-dimensional bound is left open.