Wong and Zhu's permanent-index conjecture for the total-weighting matrix

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 incidence and adjacency rule described in the source. The permanent index of a matrix $A$, denoted by ${\rm pind}(A)$, is the minimum integer $k$ for which there is a matrix formed from columns of $A$, with each column of $A$ used at most $k$ times, and with nonzero permanent.

Wong and Zhu's conjecture. For any graph $G$,

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

The source relates this permanent-index statement to $(2,2)$-choosability; its resolution is not specified.