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

Let $G$ be a graph, and let an $(a,b)$-matrix mean a square matrix $A_G(\eta)$ in which each vertex column occurs at most $a$ times and each edge column occurs at most $b$ times. Such a matrix is permanent-non-singular when its permanent is nonzero. The permanent-non-singular $(0,k)$-matrix conjecture. There is a constant $k$ such that every graph $G$ without isolated edges has a permanent-non-singular $(0,k)$-matrix. This is stated as a weaker conjecture than the preceding $(0,2)$-matrix claim, but the supplied material gives no resolution.