Harmonic-map projection conjecture for pinched Hadamard manifolds

Let $M$ be a pinched Hadamard manifold, meaning a simply connected, complete Riemannian manifold whose sectional curvature is bounded between two negative constants. Let $\iota:\mathbb{H}^2\to M$ be a quasi-isometric embedding. Harmonic-map projection conjecture. There exists a harmonic map $h:M\to\mathbb{H}^2$ such that

$$\sup_{x\in M}\operatorname{dist}(x,\iota\circ h(x))<\infty.$$

The conjecture asserts that the paper's strategy for constructing harmonic maps near nearest-point retractions works for every pinched Hadamard manifold and every such quasi-isometric embedding; the source presents it as an open conjecture following a partial positive result.