Garbit–Mustapha–Raschel conjecture on critical exponents of quadrant walks

Let ${\mathcal S}\subset\mathbb{Z}^2$ be a non-singular step set, let $Q(1,1;t)$ be the corresponding generating function, and let

$$\mathcal Q=\{(x,y)\in\mathbb{R}^2\mid x\geqslant 1\text{ and }y\geqslant 1\}.$$

Let $(x^*,y^*)$ be the unique point minimizing the characteristic polynomial $S_{\mathbf a}$ on $\mathcal Q$. Garbit–Mustapha–Raschel conjecture. The coefficient sequence $[t^k]Q(1,1;t)$ has exponential growth and its critical exponent is the one determined by $(x^*,y^*)$ according to the cases in the source's table. The source presents this as a conjecture and gives no resolution.