Universal upper bound for sharp Hölder constants

Determine whether there exist, for every integer m, an m/(m+ 1)-Hölder continuous surjection from the unit m-cube [0, 1]^m onto the unit (m+ 1)-cube [0, 1]^{m+1} whose Hölder constant is bounded above by a universal constant independent of m.

Background

The paper constructs, for every m, an m/(m+ 1)-Hölder surjection from [0, 1]^m onto [0, 1]^{m+1}, attaining the optimal possible Hölder exponent. The explicit construction in Theorem 1.1 has Hölder constant 2^{m+2}(2^m− 1), while Lemma 4.3 proves that every such surjection must have Hölder constant at least 2^{m/(m+1)}.

Question 4.4 asks whether the dependence of the known upper bound on the dimension can be replaced by a dimension-independent universal constant. Resolving it would determine whether sharp-exponent space-filling parameterizations between consecutive cube dimensions can be constructed with uniformly controlled distortion.