Has the assumption that individuals' tastes do not change over time been rigorously challenged?
Stigler and Becker's argument is methodological, not philosophical. They do not try to convince us that preferences are indeed identical across individuals and invariant across time as a matter of reality (the "Rocky Mountains" metaphor is an "as if " approach).

Their point is that any outcome can be rationalized by assuming that "it was preferences that made it so", since "De Gustibus Non Est Disputandum", and also they are unobservable.  But then, we could "explain everything" in this way, and so explain nothing.

Their goal is to defend in terms of useful modelling the other extreme: assume immovable preferences and try to find explanations for the observed outcomes based on observable, quantifiable concepts, like prices. I believe the following passage from the first page of the paper summarizes the approach


  "On the traditional view, an explanation of economic phenomena that reaches a difference in tastes between people or times is the terminus
  of the argument: the problem is abandoned at this point to whoever
  studies and explains tastes (psychologists? anthropologists?
  phrenologists'? sociobiologists?). On our preferred interpretation,
  one never reaches this impasse: the economist continues to search for
  differences in prices or incomes to explain any differences or changes
  in behavior. The choice between these two views of the role of tastes
  in economic theory must ultimately be made on the basis of their
  comparative analytical productivities."


Bold my emphasis.  

So the question of the OP appears to be mistargeted: people's tastes may very well change over time, and I don't think that Stigler and Becker would deny that. 
The question is, can we arrive at more useful economic models by assuming changing tastes, compared to the models where tastes are fixed (while avoiding the "explain everything and so explain nothing" trap)? But this would be a whole research program, not some rigorous argument in a paper.