Existence of a nice embedded tree

Let $X$ be a noncompact, metrizable generalized continuum. An infinite rooted tree is an infinite tree with a distinguished root, and a proper embedding is a topological embedding whose inverse images of compact sets are compact. The induced map on end spaces is denoted $E(\tau):E(T)\to E(X)$. Existence of a nice embedded tree conjecture. There exists an infinite rooted tree $T$ and a proper topological embedding

$$\tau:T\rightarrowtail X$$

such that

$$E(\tau):E(T)\to E(X)$$

is a homeomorphism. Moreover, $T$ may be chosen either to contain no leaves or so that the root is the unique leaf. This would provide a tree model for the end space of every noncompact metrizable generalized continuum, with additional control over the tree's leaves; the source presents it as a future direction and gives no known resolution.