The Bézout tree conjecture for Pythagorean pairs

Consider the trinary trees generated by $(2,1)$ and $(3,1)$. For a relatively prime pair $(m,n)$, let $(u,v)$ be the pair in a Bézout tree corresponding to $(m,n)$, and let $(U,V)$ be the pair given by the \texttt{\gcd} function for the same pair $(m,n)$. Bézout tree conjecture. The following hold: for every $(u,v)$ in the Bézout tree of $(3,1)$ generated by $(0,1)$, one has

$$(u,v)=(U,V).$$

One third of the $(u,v)$ in the Bézout tree of $(2,1)$ generated by $(0,1)$ are not equal to $(U,V)$. If the value of $g(0,-1)$ in the second level of this Bézout tree is changed from $(-1,2)$ to $(1,-1)$, then the resulting tree satisfies

$$(u,v)=(U,V)$$

for every $(u,v)$. These claims arise from computations of the Bézout tree to depth $13$, and describe when the tree-generated pairs agree with the pairs produced by the Euclidean algorithm; their general validity is not established in the source.