Conjecture on analytic tangent cones of admissible Hermitian–Yang–Mills connections

Let $A$ be an admissible Hermitian–Yang–Mills connection on $(\mathcal{E},B)$, and let $\hat{\mathcal{E}}$ be any chosen optimal extension of $\mathcal{E}$ at $0$. An analytic tangent-cone conjecture. There is a unique analytic tangent cone $(\mathcal{E}_\infty,A_\infty,\Sigma^{an},\mu)$ on $\mathbb{C}^n$ of $(\mathcal{E},A)$ at $0$, where $(\mathcal{E}_\infty,A_\infty)$ is a Hermitian–Yang–Mills cone, $\Sigma^{an}$ is the bubbling set, and $\mu$ is the limiting measure. Moreover,

$$\mathcal{E}_\infty\simeq\psi'_*\pi'^*((\operatorname{Gr}^{HNS}(\underline{\hat{\mathcal{E}}}))^{**}),$$ $$\Sigma^{an}=\pi'^{-1}(\operatorname{Sing}(\operatorname{Gr}^{HNS}(\underline{\hat{\mathcal{E}}}))),$$

and for each irreducible component of $\Sigma^{an}$ of pure complex codimension $2$, the analytic multiplicity assigned by $\mu$ equals the algebraic multiplicity. This conjecture predicts that the analytic tangent cone and bubbling data are uniquely determined by the algebraic optimal extension and its Harder–Narasimhan–Seshadri graded object.