Bijection conjecture for hesitating walks and excursions in the octant and quadrant

Let $n$ be a nonnegative integer. A hesitating excursion is a hesitating walk of length $2n$ in the quadrant that returns to its starting point, and a hesitating axis-walk is a hesitating walk of length $2n$ in the octant constrained to the axis. The thick diagonal is

$$\{(n,n),n\in\mathbb{N}\}\bigcup\{(n+1,n),n\in\mathbb{N}\}.$$

Bijection conjecture. The following families are in bijection: hesitating excursions of length $2n$ in the quadrant; hesitating axis-walks of length $2n$ in the octant; and hesitating walks of length $2n$ in the octant ending on the thick diagonal.

The first two families are already known to be equinumerous with Baxter numbers, while the computational enumeration mentioned in the source suggests the bijective link with hesitating walks ending on the thick diagonal. The conjecture concerns the existence of bijections among all three families.