Ranganathan–Wise differential descent conjecture for multiscale differentials

Let $(C,\eta)$ be a generalised multiscale differential with tropicalisation $\lambda$. For every level $i$, let $\lambda_i$ denote the truncation of $\lambda$ to that level, and let $\widetilde{C}$, $\tilde\eta$, $\sigma$, $\overline{C}_i$, and $\tilde\eta_i$ be as in the assertion below. Differential descent conjecture. The differential $\eta$ is smoothable if and only if: (i) for every level $i$, the truncation $\lambda_i$ is a realisable tropical differential; (ii) there exists a logarithmic modification $\widetilde{C}\rightarrow C$, a natural extension $\tilde\eta$ of the pullback of $\eta$ to $\widetilde C$, and a reduced Gorenstein contraction $\sigma\colon \widetilde C\rightarrow\overline{C}_i$ such that $\sigma^*\omega_{\overline{C}_i}=\omega_C(\lambda_i)$; and (iii) the differential $\tilde\eta_i$ at level $i$ descends to a local generator of $\omega_{\overline{C}_i}$. This is attributed in the source to D. Ranganathan and J. Wise and is presented as a conjectural algebraic characterization of smoothability; the source supplies no resolution, so it remains open.