The permanent-non-singular $(1,1)$-matrix conjecture

Let $G$ be a graph, and orient its edges arbitrarily. Define the graph matrix $A_G$ with rows indexed by $E(G)$ and columns indexed by $V(G)\cup E(G)$ by assigning to an oriented edge $e=(u,v)$ the entries $1$ in the column for $v$ and for edges other than $e$ incident with $v$, $-1$ in the column for $u$ and for edges other than $e$ incident with $u$, and $0$ elsewhere. For an index function $\eta$, let $A_G(\eta)$ repeat the column indexed by $z$ exactly $\eta(z)$ times. An $(a,b)$-matrix is a square matrix $A_G(\eta)$ with $\eta(v)\le a$ for every vertex and $\eta(e)\le b$ for every edge; it is permanent-non-singular when its permanent is nonzero. The permanent-non-singular $(1,1)$-matrix conjecture. Every graph $G$ has a permanent-non-singular $(1,1)$-matrix. This matrix formulation is motivated by total-weighting and edge-weighting problems; the stated status evidence says the conjecture is resolved for wheels and Halin graphs, while no general resolution is supplied here.