Quasi-adjunction faces and characteristic-variety components conjecture

Let $X$ be a smooth projective variety, let $D=\sum D_i$ be a divisor, and let $U=X-D$. Let $\mathcal S$ be the collection of non-normal crossings of $D$. A global face of quasi-adjunction $\mathcal F$ is a face of the intersection of the corresponding global polytopes of quasi-adjunction. If $\mathcal D\in\operatorname{Pic}X$ is a contributing divisor with $H^1(\mathcal A_{\mathcal F}\otimes\Omega_X^{n+1}\otimes\mathcal D)\ne0$, set

$$k={\rm \dim} H^1(\mathcal A_{\mathcal F}\otimes\Omega_X^{n+1}\otimes\mathcal D).$$

Quasi-adjunction characteristic-variety conjecture. The Zariski closure of $\exp(\mathcal F)\subset\operatorname{Char}H_1(U,\mathbb Z)$ is a component of the characteristic variety $V_k$.

The conjecture identifies components arising from global faces of quasi-adjunction and is supported in the paper by results for curves and for hypersurfaces with isolated singularities. The source explicitly says that it is unknown whether such components are all essential components of the characteristic variety.