Existence conjecture for harmonic extensions of admissible boundary maps

Let $X$ and $Y$ be Hadamard manifolds, and let a map between their respective boundaries at infinity be admissible. A boundary map is admissible when the inverse image of every measure-zero set has measure zero.

Harmonic extension conjecture. The admissible boundary map can be extended to a harmonic map from $X$ to $Y$ with bounded energy density if and only if the inverse image under the boundary map of every measure-zero set has measure zero.

This conjecture proposes a natural criterion for extending maps at infinity to harmonic maps of Hadamard manifolds with bounded energy density. The surrounding discussion gives examples showing that arbitrary boundary maps need not admit such extensions, but the supplied text does not state whether this conjecture has been resolved.