A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)

Question: Is it true that every (finite rank) Euclidean building has a biLipschitz embedding into a finite product of metric $\mathbb{R}$-trees?

It follows from the results of Lang-Schlichenmaier that a Euclidean building $X$ of rank $n$ admits an embedding $f$ into the product $T$ of $n+1$ trees, with the property that for some $C>1$ and $p>0$, $$ C^{-1}d_X(x,y)^p \leq d_T(f(x),f(y)) \leq Cd_X(x,y)^p.$$ So what I am asking is if there is any obstruction to taking $p=1$ (maybe increasing the number of trees in the product if necessary).

Apologies if the question is foolish, I am not an expert on buildings.