Forbidden minors for (ℓ2^d, ℓ2)-flattenability when d ≥ 4

Determine the complete finite set of forbidden graph minors that characterizes the class of graphs that are (ℓ2^d, ℓ2)-flattenable for each integer dimension d ≥ 4. Equivalently, identify all minors whose exclusion is necessary and sufficient for a graph to be (ℓ2^d, ℓ2)-flattenable when d ≥ 4.

Background

Flattenability is a minor-closed property of graphs, so by the Robertson–Seymour theorem it admits a finite forbidden minor characterization for any fixed pair of normed spaces (X, Y). In the Euclidean case, Belk and Connelly determined the forbidden minors for (ℓ2^d, ℓ2)-flattenability when d ≤ 3: K3 for d = 1, K4 for d = 2, and K5 together with K2,2,2 for d = 3.

For higher dimensions (d ≥ 4), the exact forbidden minor set is not known, although it is known to be contained within the forbidden minors of partial d-trees. Establishing the precise list for d ≥ 4 would complete the Euclidean flattenability picture across dimensions.