Le Donne's BLD embedding conjecture for compact length metric spaces

Let $(\mathsf{X},\mathsf{d}_{\mathsf{X}})$ be a compact length metric space with finite Hausdorff dimension. A bounded-length-distortion map, or BLD map, is a continuous map into a metric space whose lengths satisfy

$$\frac{1}{c}\operatorname{Len}_{\mathsf{X}}(\gamma)\leq\operatorname{Len}_{\mathsf{Y}}(\varphi\circ\gamma)\leq c\operatorname{Len}_{\mathsf{X}}(\gamma)$$

for some $c\in(0,+\infty)$ and every curve $\gamma$ in $\mathsf{X}$. Le Donne's BLD embedding conjecture. Every such $\mathsf{X}$ admits a BLD map into a Euclidean space that is also a topological embedding. The conjecture would generalize the known positive result for compact smooth manifolds equipped with sub-Finsler metrics, which are biLipschitz equivalent to sub-Riemannian metrics; its status is not resolved in the supplied source.