Necessity of the finite-dimensionality of X in the ℓ2-extremal flattenability implication

Ascertain whether the finite-dimensionality assumption on X is necessary in the following implication: if a graph G is (X, Y)-flattenable with Y infinite-dimensional, then G is (X, ℓ2)-flattenable. In other words, determine whether this implication continues to hold when X is allowed to be infinite-dimensional.

Background

The paper shows that ℓ2 and ℓ∞ act as extremal spaces for flattenability. Specifically, Theorem 4.1 establishes that if X is finite-dimensional and Y is infinite-dimensional, then (X, Y)-flattenability implies (X, ℓ2)-flattenability.

A remark immediately following the theorem notes that the requirement that Y be infinite-dimensional is essential, but leaves open whether X must be finite-dimensional, raising a structural question about the necessity of this hypothesis.