Tangent-cone conjecture for the blowdown of a projective bundle

Let $Y$ be the asymptotically conical Calabi–Yau space and let $(Z,d)$ be the metric space structure on $Z$ induced from $Y$ by the preceding convergence theorem. Set

$$V:=\mathcal{O}_{\mathbb{P}^{n-1}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{n-1}}(-(n-1)).$$

Tangent-cone conjecture. At the singular point $z\in Z$, the tangent cone to $(Z,d)$ is isometric to the blowdown of the zero section in $\mathcal{O}_{\mathbb{P}(V)}(-1)$, equipped with its conical Calabi–Yau metric. This predicts the precise local metric model at the singular point of the limiting metric space; the source gives no proof or resolution, so the conjecture remains open.