It is said in this thread Unique smooth structure on $3$-manifolds that every topological $3$-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have boundary or not and the references therein are hard to digest (since I am no expert in this field). So my question is whether every topological $3$-manifold with boundary admits a smooth structure?

By smooth structure I mean a collection of charts mapping homeomorphically onto open subsets of the (closed) upper half space, whose transition functions extend to smooth functions on some open neighbourhood of their domains.

I believe that the answer to my question is yes (assuming the theorem holds for topological $3$-manifolds without boundary) with the following reasoning:

Let $M$ be a topological $3$-manifold with non-empty boundary. Then we can consider the boundaryless double $\widetilde{M}$ of $M$, which is a topological $3$-manifold without boundary in which $M$ can be (topologically) embedded. We can now equip $\widetilde{M}$ with a smooth structure and restrict the charts to the embedding of $M$, which gives us a smooth atlas of the embedding and in turn a smooth atlas of $M$.

Is my reasoning correct?

Thanks a lot in advance!