Translated-subgroup support conjecture for homotopy characteristic varieties

Let $D$ be a divisor with isolated non-normal crossings in a sufficiently small ball $B_{\epsilon}$, and let $U=\partial B_{\epsilon}-D$. The support of the homotopy module $\pi_n(U)$ in the character torus $\operatorname{Char}\pi_1(U)$ is defined by the characters at which its associated homology support is nonzero.

Translated-subgroup support conjecture. The support of $\pi_n(\partial B_{\epsilon}-D)$ is a union of translated subgroups of $\operatorname{Char}\pi_1(\partial B_{\epsilon}-D)$ by points of finite order.

This generalizes the corresponding assertion for isolated singularities, where the support is given by the torsion eigenvalues of the Milnor monodromy. The paper presents this as a generalization of the monodromy theorem; no resolution status is supplied.