Power-law Lipschitz extension conjecture for Hadamard manifolds

Let $X$ and $Y$ be Hadamard manifolds, let $C<1$, and let $\mathcal{L}_{X,Y}(C)$ denote the optimal Lipschitz constant for extending every $C$-Lipschitz map from a subset of $X$ to $Y$. Power-law extension conjecture. There exists a universal $\alpha\in(0,1)$ such that

$$\mathcal{L}_{X,Y}(C)\leq C^\alpha.$$

This is proposed as a strengthening of the preceding uniform extension conjecture: although extension may incur loss, the ratio of the deficits from $1$ is expected not to become arbitrarily small. The source gives no resolution of this strengthening.