Uniqueness of the $[\varphi,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour

We establish a uniqueness result for the $[\varphi,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Mart\'in, J. P\'erez-Garc\'ia, A. Savas-Halilaj and K. Smoczyk proved that, if $\Sigma$ is a properly embedded translating soliton with locally bounded genus, and $\mathcal{C}^{\infty}$-asymptotic to two vertical planes outside a cylinder, then $\Sigma$ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of $[\varphi,\vec{e}_{3}]$-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.