Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:

What are the

necessaryconditions for the Freudenthal compactification of $X$ to be a metric ANR?

In the literature I have found some *sufficient* conditions (though up to date I was not able to dig through the papers I refer to and compare the conditions), like:

- forward tameness (Hughes, B., Ranicki, A.: Ends of Complexes, Chapter 7.)
- docility at infinity (Sher, R. B.: Docility at infinity and compactifications of ANR's)
- $\mathcal{C}_p$-movability at infinity (Cerin, Z.: $\mathcal{C}_p$-movable at infinity spaces, compact ANR divisors and property $UVW^n$)
- $SUV^\infty$ (Sher, R. B.: A theory of absolute proper retracts)

A full characterization of $X$'s with the property I ask for is probably difficult. An analogous question for the one-point compactification is stated open (Problem 79SC6) in the book J. van Mill, G.M. Reed: Open Problems in Topology. At least in the case of finitely many ends these two problems look at least close to being equivalent. The book has some references to the papers of Dydak ('On $LC^n$ divisors' and 'On maps preserving $LC^n$ divisors'), which probably give an answer in the finite dimensional case, but which at the moment I still do not understand (if some short exposition is possible, it would be welcome).

Perhaps for me even better than the answer to the first question would be a good answer to the following one:

What are some simple examples of $X$ whose Freudenthal compactification is not an ANR and why is it so?