This is a follow-up question to this post.
Let $G$ be a locally compact Hausdorff group, let $H$ be a closed central (i.e. $H\subset Z(G)$) subgroup and let $\iota:H\to G$ denote the inclusion. Then we get a surjective continuous map $\widehat{\iota}:\widehat{G}\to\widehat{H}$ between the unitary duals of $G$ and $H$ by assigning the central character to unirreps of $G$, where we equip $\widehat{G}$ and $\widehat{H}$ with the Fell topology (see post above).
Question: Is $\widehat{\iota}$ a quotient map?
My thoughts: I tried proving that $\widehat{\iota}$ is open or that $\widehat{\iota}$ is closed just by applying the definitions. I got stuck since I was only able to prove the necessary estimate on $K\cap H$ instead of $K$, where $K$ denotes a compact subset of $G$.
Edit: I've also tried proving the universal property of quotient maps or proving that it is a regular epimorphism (which is equivalent to being a quotient map) but this comes down to exhibiting a continuous split for $\widehat{\iota}$, which I don't think is possible.