The Gorenstein contraction conjecture for multiscale differentials

Let $(X,\eta)$ be a generalized multiscale differential with semistable modification $(\tilde{X},\tilde{\eta})$, dual graph $\tilde{\Gamma}$, levels indexed by $i$, and vanishing orders $\tilde{\ell}_i$ at those levels. Let $\tilde{X}_{>i}$ denote the union of components above level $i$, and let $\mathcal H(\mu)$ be the stratum of holomorphic differentials with zero multiplicities prescribed by $\mu$. Gorenstein contraction conjecture. (i) Suppose $(\tilde{X},\tilde{\eta})$ arises as the limit of a one-parameter family $\tilde{\mathcal C}$ of holomorphic differentials $(C_t,\omega_t)$ in $\mathcal H(\mu)$ as $t$ approaches zero. Then, for every level $i$ above all leaf vertices in $\tilde{\Gamma}$, there exists a reduced Gorenstein contraction $\sigma\colon\tilde{\mathcal C}\to\mathcal C'$ contracting exactly $\tilde{X}_{>i}$, such that $\tilde{\eta}_i$ descends to a local generator of $\omega_{X'}$, where $X'$ is the central fiber of $\mathcal C'$ and has isolated Gorenstein singularities formed by contracting each connected component of $\tilde{X}_{>i}$. Moreover,

$$\sigma^{*}\omega_{\mathcal C'}=\omega_{\tilde{\mathcal C}}\bigg(\sum_{j>i}(\tilde{\ell}_i-\tilde{\ell}_j)\tilde{X}_j\bigg).$$

(ii) Conversely, if for every level $i$ there exists a contraction $\varsigma\colon\tilde{X}\to X'$ contracting each connected component of $\tilde{X}_{>i}$ to an isolated singularity of $X'$ and $\tilde{\eta}_i$ descends to a local section of $\omega_{X'}$ at the resulting singularities, then $(X,\eta)$ can be smoothed into $\mathcal H(\mu)$. This conjecture characterizes smoothability of generalized multiscale differentials through compatible Gorenstein contractions of components above each level; the stated equivalence between one-parameter smoothings and such contractions remains unresolved here.