TITLE: Conway's subprime Fibonacci sequences
QUESTION [6 upvotes]: 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."

REPLY [4 votes]: 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)).