A Bernstein problem for translating solutions to the mean curvature flow

We study entire graphical translating solutions of the mean curvature flow, \[ \operatorname{div}\!\left(\frac{\nabla G}{\sqrt{1+|\nabla G|^2}}\right) =\frac{1}{\sqrt{1+|\nabla G|^2}} \qquad\text{in }\mathbb R^N. \] Every such graph is mean-convex, since its mean curvature is the vertical compon