Schoen's harmonic extension conjecture for hyperbolic spaces

Let $n\geq 2$, identify the ideal boundary $\partial {\mathbb {H}}^{n}$ with $\mathbb {S}^{n-1}$, and let $f:\mathbb {S}^{n-1}\to\mathbb {S}^{n-1}$ be a quasiconformal map. A map $H:\mathbb {H}^{n}\to\mathbb {H}^{n}$ is quasi-isometric if it is a quasi-isometric embedding, and it extends $f$ if its boundary map is $f$.

Schoen's conjecture. There exists a unique harmonic and quasi-isometric map

$$H:\mathbb {H}^{n}\to\mathbb {H}^{n}$$

that extends $f$.

The conjecture concerns harmonic extensions of quasiconformal boundary maps between hyperbolic spaces. The source abstract states that the conjecture is confirmed in dimension $3$, while the statement is posed for general $n$; the status outside the proved dimension is not specified here.