There are no experiments given showing that perspective lines will merge an infinite distance away, or that they will never merge.
There can’t be any such experiments. Parallel lines
by definition do not meet. I.e. they are
defined as lines which are equidistant at every point.
If, by experiment perhaps, you find a pair of lines that meet at any point, then they cannot be parallel, for the definition rules that out.
By analogy, there is no point in an experiment to find a married bachelor. If you find any married men at all, then they are not bachelors. They lost that status when they got married. Likewise, lines which join at any point have lost the status of being parallel.
[edit] Regarding railway tracks, they of course
have to be parallel, otherwise the train would come off the tracks.