Re: [SR] Their proper times will necessarily be equal

Yes, of course.
 My response differs from others because I interpret your context
differently.
 Since you mention "proper times", your context must be relativity; it
does not matter whether SR or GR, because "travel an identical path"
means they travel along a single worldline through spacetime -- i.e.
they are always co-located and co-moving, so of course their elapsed
proper times are equal (counting from any event on their worldline).
  Your "in equal observable times" is redundant. For any
observer this directly follows from them following the
same worldline through spacetime.
 Note this is essentially the first time I agree with Hachel. I doubt
that he understands why what he wrote is actually correct, because he
followed it with a bunch of obfuscatory nonsense.
 

[... enormous amount of gibberish ignored.]