Existence of a proper retract to an embedded tree

Let $X$ be a noncompact generalized continuum. Let $T$ be an infinite tree and let

$$\tau:T\rightarrowtail X$$

be a proper embedding. Assume that the induced map on end spaces

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

is injective. Existence of a proper retract to an embedded tree conjecture. There exists a proper map

$$\rho:X\to T$$

such that

$$\rho\circ\tau=\operatorname{id}:T\to T.$$

In particular,

$$\tau\circ\rho:X\to\operatorname{Im}\tau$$

is a proper retraction of $X$ onto the image of $\tau$. This is the tree analogue of the paper's proper-retract result for baserays; it is stated in the future-directions section and remains unresolved in the source.