Stable surface hyperbolicity conjecture for rank-four and rank-two forms

Let $(P,\lambda)$ be a hermitian, sesquilinear form on a free $\mathbb{Z}[\mathbb{Z}]$-module of rank $2g$, where $g\in\{1,2\}$. A form is surface hyperbolic if it is isometric to $\mathcal{H}_2^{\oplus g}$; it is stably surface hyperbolic if, for some $k$, $(P,\lambda)\oplus\mathcal{H}_2^{\oplus k}\cong\mathcal{H}_2^{\oplus g}$. The stable surface hyperbolicity conjecture. If $(P,\lambda)$ is stably surface hyperbolic, then $(P,\lambda)$ is surface hyperbolic. This algebraic conjecture is presented as implying the topological unknotting conjecture for $\mathbb{Z}$-surfaces of genus one and two; its resolution is not given in the supplied text.