Functorial correspondence between class field towers and resolved curves

Let $(0,0)$ be an isolated singular point of an algebraic curve over a number field $k$, with local ring $\mathcal{O}=k[[x,y]]/(f)$ and multiplicity $m$. Let $C$ be the nonsingular curve obtained by resolving the singularity through $m-1$ blow-ups, and let $K$ be the underlying number field. Writing $q=p^{m-1}$, denote by $\mathscr{H}^{m-1}(K)$ the $(m-1)$-fold Hilbert class field of $K$ and by $\mathbf{F}_q(C)$ the geometric extension of the rational function field defined by $C$. Functorial correspondence conjecture.

$$\mathscr{H}^{m-1}(K)\cong \mathbf{F}_{q}(C),$$

where $q=p^{m-1}$. The conjecture proposes that the correspondence between the resolved curve and the associated class field tower is functorial. Its status is not determined by the supplied source material.