From pro-$p$-Iwahori Hecke modules to $(\varphi,\Gamma)$-modules

Let $G$ be a split reductive group over ${\mathbb Q}_p$ with
connected center, let ${\mathcal H}$ denote the pro-$p$-Iwahori Hecke
algebra, with coefficients in a finite field $k$ of characteristic $p$,
corresponding to a pro-$p$-Iwahori subgroup in $G$.

Choose a maximal split torus $T$ in $G$. Depending on the choice of a
(suitable) semiinfinite reduced chamber gallery (or "alcove walk")
$C_{\bullet}=C_0,C_1,C_2,\ldots$ in the affine Coxeter complex of $G$
(w.r.t. $T$), and a suitable 'period' $\Phi\in N(T)$ of $C_{\bullet}$ of
length $\ell$ (i.e. such that $\Phi(C_i)=C_{i+\ell}$ for all $i$), there
is a functor from the category of finite
dimensional ${\mathcal H}$-modules to the category of
$(\varphi^{\ell},\Gamma)$-modules over $k((t))$.

If $G={\rm GL}_n({\mathbb Q}_p)$ we may choose $C_{\bullet}$ and $\phi$
with $\ell=1$ (this choice essentially corresponds to the choice of an
extreme simple root for $G$). Hence, composing
with Fontaine's equivalence, we obtain a functor from the category of
finite dimensional ${\mathcal H}$-modules to the category of ${\rm
Gal}_{{\mathbb Q}_p}$-representations over $k$. It induces a bijection
between simple supersingular ${\mathcal H}$-modules of dimension $n$ and
irreducible $n$-dimensional ${\rm Gal}_{{\mathbb
Q}_p}$-representations over $k$. For $n=2$, composing with the functor of
taking pro-$p$-Iwahori invariants, this is Colmez' functor $D$ (at least
on ${\rm GL}_2({\mathbb Q}_p)$-representations generated by their
pro-$p$-Iwahori invariants).