Non-homothetic complete periodic contact forms with constant Tanaka--Webster scalar curvature

We study the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature on non-compact strictly pseudoconvex CR manifolds. We prove that, under mild assumptions, the universal cover of a compact strictly pseudoconvex CR manifold admits infinitely many non-homothetic such contact forms whenever its fundamental group has infinite profinite completion. As applications, we treat complements of real or complex spheres in the standard CR sphere, as well as circle bundles over compact K\"{a}hler manifolds and the boundary of a Reinhardt domain.