TITLE: Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2
QUESTION [5 upvotes]: In this recent MOF question I asked whether exact groups could be characterized via the existence of amenable actions on unital C*-algebras.  The answer, provided by Caleb Eckhardt in a comment, was that an amenable action on a unital C*-algebra leads to an amenable action on the compact spectrum of the center of said algebra, and hence exactness follows from a well known result.
The present question is pretty similar to my previous question, but I am now employing a different amenability condition, as follows:
Definition.  An action $\theta$ of a discrete group $G$ on a C*-algebra $A$ is said to satisfy the approximation property if there exists a net
  $\{ a_i \}_{i\in I}$
  of finitely supported functions
  $$
  a_i: G \to A,
  $$
  which is bounded in the sense that
  $$
  \sup _{i\in I} \Big\Vert {\sum_{g\in G } a_i(g)^* a_i(g)} \Big\Vert < \infty ,
  $$
  and such that
  $$
  \lim _{i \rightarrow \infty } \sum_{h\in G } a_i(gh)^* b\,\theta_ g( a_i(h)) = b,\quad \forall b\in A .
  $$
See the discussion before [1, Definition 20.11].
This condition is weaker than Anantharaman-Delaroche's usual condition of amenability (definition 4.3.1 in Brown and Ozawa) in that it does not require the $a_i$ to take values in the center of $A$.  Nevertheless it is enough for most purposes and in particular it implies that the crossed product is nuclear provided $A$ is nuclear [1, Proposition 25.10].  
Thus, here is the new version of my question: 

Question: Suppose that a discrete group $G$ admits an action on a unital
  C*-algebra $A$, satisfying the above approximation property.  Is $G$
  necessarily exact?

Notice that, precisely because the $a_i$ are allowed to take values outside the center of $A$, it is not immediately clear that the above condition passes to the center, and hence Caleb's answer might no longer work in this case.
Reference:
[1]
Partial Dynamical Systems Fell Bundles and
Applications

REPLY [3 votes]: The answer to the question is positive: see Remark 6.6 in the paper
https://arxiv.org/pdf/1904.06771.pdf
The approximation property implies amenability in the sense of Claire Anantharaman Delaroche, this has been proved in
https://arxiv.org/pdf/1907.03803.pdf
for discrete groups, see also
https://arxiv.org/pdf/1904.06771.pdf
Indeed, it has been stablished recently that the approximation property is equivalent to amenability even in the more general context of locally compact groups by Ozawa-Suzuki in
https://arxiv.org/abs/2011.03420
and that the above question in this context also has a positive answer (see Corollary 3.6).