I do have two vague questions, though. Did the inspiration for $Y$ come only from the example of Borsuk-Mazurkiewicz? Any advice on how to find relevant examples like these in the old literature on geometric topology? Thanks a lot for your amazing answers. I may eventually mark this answer as the accepted one.

[continuation of comment] 2) The part about collapsing $D^2\times [0,1]$ mapping onto $Q$ does not seem to work unless you replace $Q$ by $Q_2$, and only after implementing the change I mentioned in (1). 3) The numbers seem to be a little off in the last paragraph. Sorry about the nitpicking. I am just trying to make sure I understand the example correctly by working out the kinks.

@Sergey: Truly great work! Thank you so much. I am still thinking about the details. In the meantime, just a few remarks. 1) If I understand the example correctly, to make $Q_k$ contractible, it seems that you want to start with $(Q_1,\partial Q_1)=(D^2,\partial D^2)$ (instead of $(Q,\partial Q)$). This change doesn't make much difference other than ensuring that $Q_k$ and $Q_\infty$ are contractible, and either possibility seems to give an answer to my question.

@Sergey: Many thanks for the edit and answer to my comment. Is it then true that the image of $\{\infty\}\times\Bbb{R}$ in $S^3\times\Bbb{R}$ cannot be locally flat at any point?

This is a very interesting answer. Absolutely not what I had in mind. I am still wading through the details. I do have a question, though. Do you know if the homeomorphism $f:W^+ \times \Bbb{R}\rightarrow S^3\times \Bbb{R}$ can be chosen so that $f(\{\infty\}\times\Bbb{R})=\{p\}\times\Bbb{R}$?

@Mike: Unfortunately, the deformation of the neighbourhood to the base-point does not have to remain within the neighbourhood. Thus, the neighbourhood need not be contractible.