Tangent-cone conjecture for partial resolutions of $Y_{p,p}$

Let $1\leqslant k<p-1$, and let $\overline{Y}$ be the partial resolution obtained by blowing up $k$ of the lines $x=z-\zeta^jw=0$. Its isolated singularity is modeled by

$$\widetilde{Y}_{p-k,p-k}:=\{xy=\prod_{j=k}^{p-1}(z-\zeta^jw)\}\subset\mathbb{C}^4.$$

Let $d$ be the metric completion of $(\overline{Y}_{\rm reg},\bar{\omega})$. Tangent-cone conjecture. The tangent cone to $(\overline{Y},d)$ at the singular point is isometric to $\widetilde{Y}_{p-k,p-k}$ equipped with its conical Calabi–Yau metric. This predicts that the metric singularity is asymptotic to the conical Calabi–Yau metric on the corresponding deformation-equivalent isolated singularity; the source presents this as an expectation, and no resolution is given.