Euclidean dimension bounded by VC-dimension for ideals of convex geometries

Let $\mathcal{I}=(U,\mathcal{C'})$ be an ideal of a convex geometry, with Euclidean dimension $\operatorname{\dim_{\mathbb{E}}}(\mathcal{I})$ and VC-dimension $\operatorname{\dim_{VC}}(\mathcal{I})$. Euclidean-dimension bound. The Euclidean dimension $\operatorname{\dim_{\mathbb{E}}}(\mathcal{I})$ is always upper bounded by a function of its VC-dimension $\operatorname{\dim_{VC}}(\mathcal{I})$. The corollary preceding this claim establishes the equality of these dimensions for distributive lattices, while the proposed bound for arbitrary ideals remains open.