Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and $\tau_{y, w}$ be the parallel transports of the tangent planes at $x$ and $y$ to the tangent plane at $w$, respectively. Does there exist a positive constant $C$ such that
$$\|\tau_{x,w}\circ Log_{x}(z) - \tau_{y,w}\circ Log_{y}(z)\|\leq C d(x,y),$$ where $C$ is independent of $x, y, z$?
Thanks in advance!