The distance-three vertex conjecture for Pythagorean walks

Let $\mathcal{G}$ be the graph of Pythagorean walks, with distinguished vertex $O(0,0)$, and let graph distance be measured from $O(0,0)$. A vertex is a symmetric counterpart of a listed vertex if it is obtained by the symmetries of the graph used in the paper. Distance-three vertex conjecture. The only vertices at distance $3$ from $O(0,0)$ are $(1,0)$, $(2,0)$, $(2,1)$ and their symmetric counterparts. The conjecture is motivated by a computer search covering all nodes $(g,h)$ with $0\leq g,h<4000$, where all non-trivial cases were found to have distance at most $2$; the existence of further vertices requiring distance $3$ or more therefore remains open.