Intrinsicness conjecture for weighted tangent cones and their filtrations

Let $p$ be a germ of an analytic singularity, and let $W$ and $C(Y)$ be the polarized affine varieties arising as its weighted tangent cone and metric tangent cone, respectively. Let the filtration in question be the filtration associated with equation $(4.3)$.

Intrinsicness conjecture. The filtration

$$(4.3)$$

and the polarized affine varieties $W$ and $C(Y)$ are uniquely determined by the germ of the analytic singularity $p$. In particular, they are independent of the Kähler–Einstein metric on $Z$ defining them.

The conjecture asserts that these constructions are intrinsic algebraic-analytic invariants of the singularity, rather than artifacts of the chosen local embedding or Kähler–Einstein metric. The source provides no evidence of a resolution or refutation.