Let $\alpha\subset\Pi\subset M$ be geodesic $\alpha$ contained in a geodesic plane $\Pi$ in a $\operatorname{CAT}(0)$ space $M$, and for any convex geodesic subspace $X$ let $p_X:M\to X$ be the projection $\pi_X(p)=\{q\in X|\operatorname{dist}(p,q)=\operatorname{dist}(p,X)\}$.
Can i first project to the plane and then to the line? i.e. is it true that $p_\alpha\circ p_\Pi=p_\alpha$?
This is true for euclidean spaces, but I don't see it even for e.g. hyperbolic spaces