Let $H$ be a discrete group, and let $X$ be the one-point compactification of $\mathbb{N}$. Consider the étale groupoid $G = H \times \{\infty\} \sqcup \mathbb{N}$, whose unit space is $X$, and with operations:

1. $G$ is a group bundle: $s(g) = r(g)$ for every $g \in G$
2. The isotropy at every $k \in \mathbb{N}$ is trivial: $G_k = \{g \in G \mid s(g) = k\} = \{k\}$
3. The isotropy at $\infty$ is $H$: $G_\infty = \{g \in G \mid s(g) = \infty\} = H$

Question: can $C_r^*(G)$ be quasi-diagonal, and $H$ be non-amenable?

Note that (modulo the Tikuisis-White-Winter theorem) it is not hard to prove that if $H$ is amenable then $C_r^*(G)$ is quasi-diagonal. Moreover, a non-amenable fiber can only happen in a non-isolated point of $X$.

Recall: we say that a separable C*-algebra $A$ is quasi-diagonal if there are contractive and completely positive maps $\varphi_n \colon A \rightarrow M_{k(n)}$ such that $||\varphi_n(a)|| \rightarrow ||a||$ and $||\varphi_n(ab) - \varphi_n(a)\varphi_n(b)|| \rightarrow 0$ for every $a, b \in A$

Let $H$ be a discrete group, and let $X$ be the one-point compactification of $\mathbb{N}$. Consider the étale groupoid $G = H \times \{\infty\} \sqcup \mathbb{N}$, whose unit space is $X$, and with operations:

1. $G$ is a group bundle: $s(g) = r(g)$ for every $g \in G$
2. The isotropy at every $k \in \mathbb{N}$ is trivial: $G_k = \{g \in G \mid s(g) = k\} = \{k\}$
3. The isotropy at $\infty$ is $H$: $G_\infty = \{g \in G \mid s(g) = \infty\} = H$

Question: Can $C_r^*(G)$ be quasi-diagonal, and $H$ be non-amenable?

Note that (modulo the Tikuisis-White-Winter theorem) it is not hard to prove that if $H$ is amenable then $C_r^*(G)$ is quasi-diagonal. Moreover, a non-amenable fiber can only happen in a non-isolated point of $X$.

Recall: We say that a separable $C^*$-algebra $A$ is quasi-diagonal if there are contractive and completely positive maps $\varphi_n \colon A \rightarrow M_{k(n)}$ such that $\|\varphi_n(a)\| \rightarrow \|a\|$ and $\|\varphi_n(ab) - \varphi_n(a)\varphi_n(b)\| \rightarrow 0$ for every $a, b \in A$.