Transcendence of generating functions of walks on the slit plane

Consider a single walker on the slit plane, that is, the square grid Z^2 without its negative x-axis, who starts at the origin and takes his steps from a given set S. Mireille Bousquet-Melou conjectured that – excluding pathological cases – the generating function counting the number of possible walks is algebraic if and only if the walker cannot cross the negative x-axis without touching it. In this paper we prove a special case of her conjecture.