Cubical approximation conjecture

Let $P$ and $Q$ be two cubes, and let $f:V(P)\rightarrow V(Q)$. A cubical map is a map between the corresponding cubical face complexes that sends each cube into a cube. Cubical approximation conjecture. There exists $N\geq 0$ and a cubical map

$$g:\mathsf{csd}^N P\rightarrow Q$$

such that for each $v\in V(P)$ one has $f(v)=g(v)$, where $Q$ is identified with its face complex. Here $\mathsf{csd}$ denotes the cubical subdivision, obtained by taking the Cartesian product of the barycentric subdivisions of the one-dimensional cubes whose product defines the cube. This is proposed as a first step toward a cubical analogue of the simplicial approximation theorem; the source does not provide a resolution.