Finitary version of the dependence theorem

Determine whether, for every pair of positive integers d and k, there exists an integer N=N(d,k) such that, for every set X and every function f:X^d\to X, the absence of partitions d=u\sqcup v and tuples x_m\in X^u, y_m\in X^v satisfying f(x_l^\frown y_l)\ne f(x_m^\frown y_n) for all l<N and m<n<N implies the existence of a partition X=\bigsqcup_{j<k}U_j for which, on every corresponding rectangle determined by s\in k^d, the function f depends on at most one coordinate.

Background

The paper proves an infinitary dichotomy for functions on finite-dimensional combinatorial cubes. Given f on a product of d sets, either the domain admits a finite rectangular partition on each cell of which f depends on at most one coordinate, or there are sequences of partial tuples witnessing a strong failure of one-coordinate dependence through infinitely many inequalities.

The unresolved question asks whether this dichotomy has a uniform finitary analogue. Specifically, it asks whether the existence of sufficiently large finite witnesses to the second alternative can be bounded solely in terms of the dimension d and the number k of pieces in the partition. If no such bounded witness exists, the desired conclusion would be a k-piece partition whose every cell exhibits one-coordinate dependence.