Uniform Lipschitz extension conjecture for Hadamard manifolds

Let $C<1$. For Hadamard manifolds $X$ and $Y$ satisfying

$$\kappa_X\geq -1\geq \kappa_Y,$$

write $\mathcal{L}_{X,Y}(C)$ for the optimal Lipschitz constant needed to extend every $C$-Lipschitz map from a subset of $X$ to $Y$. Uniform Lipschitz extension conjecture. For every $C<1$, there exists $C'\in(C,1)$ such that

$$\mathcal{L}_{X,Y}(C)\leq C'<1.$$

The conjecture asserts that the lower curvature bound on $X$ and the upper curvature bound on $Y$, as well as the dimension bounds in the paper's main theorem, are not necessary. The source further notes that an equivariant version should hold, while loss of Lipschitz constant can occur in general.