Existence of weak tangent fields without an exceptional set

Determine whether, for integers m ≥ n > 2, every metric space X with H^n(X) = 0 admits an (n−1)-dimensional weak tangent field with respect to every Lipschitz mapping φ: X → R^m. Equivalently, establish whether the exceptional null set N in Theorem 1.9 (which ensures an (n−1)-dimensional weak tangent field on S \ N for purely n-unrectifiable sets S) can always be taken to be empty. The general case reduces to the Euclidean case with X = R^n, m = n, and φ equal to the identity; this Euclidean case is known for n = 2 and is announced for n ≥ 3.

Background

The paper proves Theorem 1.9 (stated as Theorem 3 in the introduction) which shows that for a Lipschitz map φ: X → R^m and a purely n-unrectifiable set S with finite H^n-measure, one can remove a null set N so that S \ N possesses an (n−1)-dimensional weak tangent field with respect to φ. A weak tangent field assigns to each point a subspace of R^m such that the tangents of φ ◦ γ lie in the assigned subspace for almost every parameter along curve fragments γ.

The authors note that a natural strengthening would eliminate the exceptional set N entirely, asserting that every H^n-null metric space has an (n−1)-dimensional weak tangent field with respect to every Lipschitz φ: X → R^m. They explain that this general question reduces to the Euclidean case X = R^n with φ = Id. This Euclidean case is known when n = 2 by results of Alberti, Csörnyei, and Preiss, and the case n ≥ 3 is expected to follow from announced results of Csörnyei and Jones.