The p-harmonic boundary extension conjecture

Let $1 < p \in \mathbb{R}$ and let $X$ be a metric measure space. Let $\Delta_p(X)$ denote the $p$-harmonic boundary of $X$, and let $BD^p(X)$ and $HBD^p(X)$ denote the bounded Royden algebra and bounded $p$-harmonic Royden algebra, respectively. For $x \in \Delta_p(X)$, let $(x_n)$ be a sequence in $X$ converging to $x$. The p-harmonic boundary extension conjecture. If $f$ is a continuous function on $\Delta_p(X)$, then there exists a $p$-harmonic function $h$ on $X$ such that

$$\lim_{n \rightarrow \infty} h(x_n) = f(x).$$

Moreover, if $f \in BD^p(X)$, then $h \in HBD^p(X)$ and $h=f$ on $\Delta_p(X)$. The conjecture proposes an extension result without the $(p,p)$-Sobolev inequality assumed earlier in the paper; the source gives no evidence that it has been resolved.