Normalization of tropicalizations of singular curves

Let $C$ be an irreducible affine algebraic curve with one singular point. Suppose $C$ is embedded with ideal $I\subseteq K[x_1,\ldots,x_n]$, and set

$$A=S[x_1,\ldots,x_n]/\operatorname{Bend}(\operatorname{trop} I),\qquad R=\operatorname{Frac}(A).$$

Let $\overline{A}^{\mathrm{V}}$ denote the valuative integral closure of $A$ in $R$, and let $B=K[x_1,\ldots,x_n]/I$.

Normalization conjecture for tropical curves. There is an embedding of $C$ with an ideal $I$ having the property that the integral closure of $B$ in its field of fractions is obtained by adjoining to $B$ lifts of the elements of $\overline{A}^{\mathrm{V}}\backslash A$.

This predicts that, for a suitable embedding, normalization of the algebraic curve can be recovered from the valuative integral closure of the corresponding tropical coordinate semiring. The examples discussed in the source support the claim for tropicalizations of curves with a unique singular point at the origin, but the general assertion remains open.