Lipeomorphic invariance of metric-coordinate dimension

Let $(M,d)$ and $(N,e)$ be metric spaces. A lipeomorphism between them is a bijective Lipschitz map whose inverse is also Lipschitz. The metric-coordinate dimension of a metric space is the smallest cardinal number $I$ such that every point has a neighborhood metrically coordinatized by a set of cardinality $I$.

Lipeomorphic invariance conjecture. Metric-coordinate dimension is invariant under lipeomorphisms; that is, if $M$ and $N$ are lipeomorphic, then they have the same metric-coordinate dimension.

This would show that metric-coordinate dimension, although not a homeomorphic invariant, is preserved by the stronger equivalence relation of lipeomorphism. The source states the claim without providing evidence of a proof or disproof.