The Bernstein problem for $(X,Y)$-Lipschitz surfaces in three-dimensional sub-Finsler Heisenberg groups

We prove that in the Heisenberg group ${\mathbb{H}}^1$ with a sub-Finsler structure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions.