I asked this question, with the hopes someone might help me with the algebraic proof I was working on towards the stated conjecture. Based on this meta thread, my original question got changed — for the better, at least from the perspective of acceptance on MO. (Based on the answers and ongoing thread, I think it's safe to say the question was indeed of sufficient “research-level” for this site.)
Despite the “success” of the thread as it stands, I'm still very interested in obtaining assistance with the algebraic proof (i.e., my original question), since I still believe the conjecture to be true, and my Vieta-jumping approach is the most successful approach I’ve found so far towards a solution.
Hence my current [meta] question: Since the existing thread has become primarily a search for counterexamples via computation, should I start a new thread with regard to completing or improving my original algebraic proof/approach? Should I post a partial answer on the current thread, to trigger a separate “sub-thread” focusing on the algebraic proof? Or should I edit my original question, to “bump” it?