Bounded-length-distortion embedding conjecture for compact length metric spaces

Let $X$ be a compact length metric space of finite Hausdorff dimension. A map from $X$ into a Euclidean space has bounded length distortion if it distorts lengths by a controlled ratio. BLD embeddings. $X$ can be embedded in some Euclidean space via a map with bounded length distortion. This would extend the known result for sub-Finsler manifolds, which are locally biLipschitz equivalent to subsets of Euclidean spaces with the path distance; the conjecture concerns arbitrary compact length metric spaces of finite Hausdorff dimension.