Tangent cone of a proper CAT(0) is a proper CAT(0) space

Question

Let $(X,d)$ be a proper CAT$(0)$ space. Let $x\in X$ and let $T_x X$ be the tangent cone of $X$ at $x$ equipped with its usual distance denoted $d_x$. It is a known fact that $(T_x X, d_x)$ is a complete CAT$(0)$ space. My question is whether it is also a proper space since $X$ is a proper space.

Answer 1

No, it need not be proper.

Indeed, let $T$ be a real tree consisting of one vertex $v$ with a sequence $(e_n)$ of edges of length $1/n$. Then $T$ is compact.

But the tangent cone of $T$ at $v$ is a real tree in which there are, for every $r>0$, infinitely many points at distance $r$ from $v$, pairwise at distance $2r$. So it is not proper, and not even locally compact.

Answer 2

The answer is "yes" for geodesically complete CAT(к) spaces. It follows directly from the comparison.