Wong–Zhu's permanent-index conjecture for $A_G$

Let $G$ be a graph, and let $A_G$ be the matrix whose rows are indexed by the edges of $G$ and whose columns are indexed by the vertices and edges of $G$, with entries determined by the oriented graph polynomial. For a matrix $A$, its permanent index $\operatorname{pind}(A)$ is the least integer $k$ for which a square matrix formed by repeating columns of $A$, with each column used at most $k$ times, has nonzero permanent. Wong–Zhu's permanent-index conjecture.

$$\operatorname{pind}(A_G)=1.$$

If true, this would imply that every graph is $(2,2)$-choosable. The conjecture is the permanent-index counterpart of the $(2,2)$ list total-weighting assertion, and remains unresolved in the supplied text.