$$ \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mbb}[1]{\mathbb{#1}} \newcommand{\opn}[1]{\operatorname{#1}} \DeclareMathOperator\cap{cap} \def\sse{\subseteq} $$
Coarse spaces
Let $X$ be a coarse space, i.e. a set equipped with a subset $\mathcal{E}$ of the powerset $\wp(X\times X)$ such that $\mathcal{E}$ is closed under inverses, finite unions, products (compositions), subsets, and contains the diagonal. The set $ \mc{E}$ is called coarse structure or the set of entourages.
Capacity with respect to an entourage
Define the capacity of a subset $S\subseteq X$ with respect to symmetric subset $E\subseteq X\times X$ as the largest number $m$ for which there exist distinct points $y_1,\ldots,y_m\in S$ such that no pair $(y_i,y_j)_{i\neq j}$ belongs to $E.$ We denote capacity of $S$ w.r.t. $E$ as $\operatorname{cap}_E(S).$
$E$-balls
For a subset $E\subseteq X\times X$ and a point $x\in X$ denote by $E_x$ the $E$-ball centered at $x,$ that is:$$ E_x=\{x'\in E\mid (x',x)\in E\}. $$
Universal bounded geometry coarse structures
We call a symmetric subset $E\subseteq X\times X$ containing the diagonal gauge. A subset $F\subseteq X\times X$ is called uniform w.r.t. $E$ if the values of $\cap_E(F[E_x])$ and $\cap_E(F^{-1}[E_x])$ are uniformly bounded in $x.$
For a fixed gauge $E$ the set of all uniform subsets in $X\times X$ forms a coarse structure Roe, Proposition 3.7. Such coarse structure is called universal bounded geometry coarse structure.
A coarse structure $\mc{E}$ has bounded geometry it is finer than universal bounded geometry coarse structure for some gauge $E$ on $X.$
Growth functions
Let $(X,\mc{E})$ be a coarse space. Assume that $\mc{E}$ has bounded geometry finer than universal geometry defined by gauge $E$. For $F\in \mathcal{E}$ define the growth function of $F$ at a point $x\in X$ as the capacity: $$ \operatorname{gr}_{F,x}:n\mapsto \operatorname{cap}_E((F^n)_x) $$
We say that the coarse space $(X,\mc{E})$ has growth type $f$ for some $f:\mbb{N}\to \mbb{N}$ if for every $F\sse X\times X$ we have $\opn{gr}_{F,x}\preceq f$ with respect to the usual (in geometric group theory) order relation on such functions and there is at least one pair $F,x$ such that $f\preceq \opn{gr}_{F,x}.$
Homotopy theory
There is a way to define the homotopy theory of coarse spaces and coarse maps using an appropriate coarse interval, see e.g. [Norouzizadeh Chapter 3] for this construction. Another more high-brow way of doing the same thing is given (if I understand correctly) by a motivic construction of Bunke and Engel, Section I.3.
The question
Is there a way to recover the growth function of a coarse space with bounded geometry in the sense of Roe, Chapter 3 from coarse homotopy invariants of this space (in the sense of Bunke and Engel or Norouzizadeh)?
EDIT: It was pointed out in the comments that I should explain all the definitions from the question for it to make any sense. I did that to the best of my abilities, although maybe the result became too lengthy.
