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." }