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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?

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    $\begingroup$ Related: meta.mathoverflow.net/questions/1845/… $\endgroup$ – Eric Wofsey Sep 6 '14 at 1:49
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    $\begingroup$ Several highly knowledgeable users seem to agree that the conjecture is very likely false, and are in the process of proving this. If you believe in your approach go ahead and wrok on it. To ask another question seems inappropriate to me. $\endgroup$ – user9072 Sep 6 '14 at 10:42

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