The permanent-non-singular $(0,2)$-matrix conjecture

Let $G$ be a graph, and define the graph matrix $A_G$ and the matrices $A_G(\eta)$ as above. 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 $(0,2)$-matrix conjecture. Every graph $G$ without isolated edges has a permanent-non-singular $(0,2)$-matrix. The claim is the matrix counterpart of the 1-2-3 conjecture, and the supplied status evidence records proofs for wheels and Halin graphs rather than a general solution.