The symmetry conjecture for exponential growth of quarter-plane walks

Let ${\mathfrak{S}}$ be a step set symmetric about an axis $L$, where $L$ passes through the first quadrant, potentially along a boundary. Let $L^\perp$ be the line perpendicular to $L$, and consider walks in the half plane bounded by $L^\perp$. The symmetry conjecture. The half-plane walks bounded by $L^\perp$ should have the same exponential growth as the quarter-plane walks on ${\mathfrak{S}}$. This conjecture concerns the use of symmetry to obtain sharp upper bounds for quarter-plane models of arbitrary step size; experimental evidence motivates it, but the general assertion remains open.