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I want to be certain I have the latest information on Conway's subprime Fibonacci sequences, arXiv-posted a year ago; I am referencing the status in a review. To wit, starting with $(0,1)$:1 $$ 0, 1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, 15, 23, 19, \ldots $$

Richard K. Guy, Tanya Khovanova, Julian Salazar:
"however, it seems more likely here than in the $3x + 1$ problem that sequences do not increase indefinitely. Here is an informal argument that supports such a conjecture,"

$\color{Red}{Q}$: Has it been established that some sequence increases indefinitely? Or no sequence, regardless of starting conditions, does not?

1"Start with the Fibonacci sequence 0, 1, 1, 2, 3, 5, . . . , but before you write down a composite term, divide it by its least prime factor so that this next term is not 8, but rather 8/2 = 4."

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    $\begingroup$ Also published as Conway’s Subprime Fibonacci Sequences, Richard K. Guy, Tanya Khovanova and Julian Salazar, Mathematics Magazine Vol. 87, No. 5 (December 2014), pp. 323-337. $\endgroup$
    – Gerry Myerson
    Commented Aug 7, 2015 at 23:06

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A somewhat different (but related) sequence has been analyzed satisfactorily. Quite recently, too (A Note on Prime Fibonacci Sequences, Jeremy Alm, Taylor Herald (Submitted on 17 Jul 2015)).

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