An old conjecture says that, for the two-dimensional system of ordinary differential equations \(\dot x=f(x)\), where \(f:\mathbb{R}^ 2\to\mathbb{R}^ 2\), \(f\in C^ 1\), and \(f(0)=0\), the origin \(x=0\) should be globally asymptotically stable (i.e., a stable equilibrium and all trajectories \(x(t)\) converge to it as \(t\to+\infty)\) whenever the following conditions on the Jacobian matrix \(J(x)\) of \(f\) hold: \(tr J(x)0\), \(\forall x\to\mathbb{R}^ 2\). It is known that if such an \(f\) is globally one-to-one as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the system \(\dot x=f(x)\). In this paper we outline a new strategy to tackle the injective of \(f\), based on an auxiliary boundary value problem. The strategy is shown to be successful if the norm of the matrix \(J(x)^ TJ(x)/\text{det} J(x)\) is bounded or, at least, grows slowly (for instance, linearly) as \(| x|\to+\infty\).