Least gradient problem with respect to a non-strictly convex norm

We study the planar least gradient problem with respect to an anisotropic norm $\varphi$ for continuous boundary data. We prove existence of minimizers for strictly convex domains $\mathrm{\Omega }$. Furthermore, we inspect the issue of uniqueness and regularity of minimizers only in terms of the modes of convexity of $\varphi$ and $\mathrm{\Omega }$. The results are independent from the regularity of $\varphi$.