It is a well-known theorem first proved by Bourgain that any map $\varphi:T_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a universal constant. Distortion is the least $D$ such that for some $s$ and all $x,y\in T_n$, we have $$ s d(x,y) \leq d(\varphi(x),\varphi(y)) \leq sD d(x,y)$$ (This also generalizes in some way to many more Banach spaces).

Some of the proofs use the linear structure of $H$, but others are purely metric. I therefore wonder:

Has this distortion estimate been generalized to spaces of non-negative curvature (Alexandrov spaces)?