Bousquet-Mélou's algebraicity conjecture for slit-plane walk generating functions

Let $\mathfrak S\subset\mathbb Z^2$ be a finite set of steps. For walks on the slit plane starting at the origin, define the generating function counted by length and final coordinates by

$$S(x,y;t)=\sum_{\substack{\text{$W$ walk on the slit plane}\\ \text{starting at the origin}\\ \text{with steps in $\mathfrak S$}}} t^{\operatorname{length} W}x^{\operatorname{x-final} W}y^{\operatorname{y-final} W}.$$

Suppose that the set of steps is not degenerated, so that all four quadrants of the plane can be reached by some walk, and that the greatest common divisor of the vertical parts of the steps is equal to one. Bousquet-Mélou's conjecture. The generating function $S(x,y;t)$ is algebraic in $t$ if and only if the height of any step is at most one.

The claim characterizes algebraic generating functions for a broad class of slit-plane walks; the paper proves a special case, while the full assertion is presented as a conjecture.