  We prove that for any possibly-punctured surface with non-empty boundary
$\mathbf{\Sigma}=(\Sigma, \mathbb{M}, \mathbb{P})$, and any tagged
triangulation $T$ of $\mathbf{\Sigma}$ in the sense of
Fomin--Shapiro--Thurston, the coefficient-free bangle functions of
Musiker--Schiffler--Williams coincide with the coefficient-free generic
Caldero--Chapoton functions arising from the Jacobian algebra of the quiver
with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the
second author.
  When the set of boundary marked points $\mathbb{M}$ has at least two
elements, Schr\"oer and the first two authors have shown, relying heavily on
results of Mills, Muller and Qin, that the generic coefficient-free
Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster
algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$. So, the
set of bangle functions proposed by Musiker--Schiffler--Williams over ten years
ago is indeed a basis.
