The torsion-translated subtori conjecture for characteristic varieties

Let $X$ be the singularity under consideration, and let $V_k$ denote the characteristic variety of $\pi_n(\partial B_{\epsilon}-\partial B_{\epsilon}\cap X)\otimes {\bf C}$. The character torus is

$$\operatorname{Spec}{\bf C}[\pi_1(\partial B_{\epsilon}-\partial B_{\epsilon}\cap X)].$$

Torsion-translation conjecture. The characteristic variety is a union of translated subtori of

$$\operatorname{Spec}{\bf C}[\pi_1(\partial B_{\epsilon}-\partial B_{\epsilon}\cap X)],$$

with each translation given by a point of finite order.

This conjecture proposes that the characteristic variety has the same structural form as in the case of curves: its components should be subtori translated by torsion characters. The source explicitly states that the equality between the union of principal components and the characteristic variety is unknown, and gives this statement as a conjecture.