Directional movement of a collective of compassless automata on square lattice of width 2

We study the following problem: Can a collective of finite automata maintain directed movement on a two-dimensional integer lattice of width 2, where the elements (vertices) are anonymous? The automata do not distinguish between vertices based on their coordinates of direction (that means each automaton has no compass). We considered collectives consisting of an automaton and some pebbles, which are automata of the simplest form, whose positions are entirely determined by the automaton. We demonstrate that a collective of one automaton and a maximum of three pebbles cannot maintain a direction of movement on the lattice. However, a collective of one automaton and four pebbles can do so.