Let $G$ be an étale locally compact Hausdorff groupoid (possibly second-countable) and let $a\in C_{\textrm{red}}^*(G)$. Is it true that for all $\varepsilon>0$ there is $s\in C_c(G)$ satisfying the following two conditions?

(1) $\|a-s\|_{\textrm{red}}<\varepsilon$

(2) $\textrm{supp}(s)\subset\textrm{supp}(a)$

(This is true in particular situations, for instance if $a$ itself is compactly supported.)

Let $G$ be an étale locally compact Hausdorff groupoid (possibly second-countable) and let $a\in C_{\textrm{red}}^*(G)$. Is it true that for all $\varepsilon>0$ there is $s\in C_c(G)$ satisfying the following two conditions?

(1) $\|a-s\|_{\textrm{red}}<\varepsilon$

(2) $\textrm{supp}(s)\subset\textrm{supp}(a)$

(This is true in particular situations, for instance if $a$ is supported in an open bisection.)