TITLE: Equi-Hölder embeddings of compact metric spaces of finite packing dimension into $\ell_2$
QUESTION [8 upvotes]: Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space?
A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder embedding if there are positive real constants $c,C,\alpha$ such that 
$$c\cdot d_X(x,y)^\alpha\le d_Y(f(x),f(y))\le C\cdot d_X(x,y)^\alpha$$
for all $x,y\in X$. 
The packing dimension of a compact metric space $(X,d)$ in the (finite or infinite) number
$$Dim(X)=\limsup_{\varepsilon\to 0}\frac{\ln N_\varepsilon(X)}{\ln(1/\varepsilon)},$$
where $N_\varepsilon(X)$ is the cardinality of the smallest cover of $X$ by subsets of diameter $\le\varepsilon$.
Remark. By the Assouad Embedding Theorem, a metric space $X$ admits an equi-Hölder embedding in a finite-dimensional Hilbert space if and only if $X$ is doubling. It can be shown that doubling metric spaces have finite packing dimension.

REPLY [3 votes]: Not an answer, but related:
In [1] (see also [2]) the authors prove the following:
If $X$ has packing dimension (in your sense; they call it "fractal dimension") less than $m/2$, then $X$ has an embedding $f$ into $\mathbb{R}^m$ such that
$$ L d(x,y)^\gamma \leq |f(x)-f(y)| \leq d(x,y) $$
for all $x,y\in X$, where $L>0$ and $\gamma\geq 1$. 
This is not the same as what you ask, but it is the only result of this type that I know that assumes finite packing dimension rather than finite Assouad dimension.
[1] C. Foias, E. Olson, E. Finite fractal dimension and Hölder-Lipschitz parametrization. Indiana Univ. Math. J. 45 (1996), no. 3, 603–616.
[2] J. Robinson, 
Dimensions, embeddings, and attractors. 
Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.