Divergent $3n+1$ sequence?

Question

Recall the Collatz function given by: $$ T(n) = \begin{cases} {\dfrac{n}{2}} & n \equiv 0\pmod 2\\ & \\ 3n+1 & n \equiv 1\pmod 2 \end{cases} $$ The well-known conjecture states that $T^{(k)}(n)=1$ for all $n$, and $k$ large enough.

Is it correct that this is not known that $T^{(k)}(n)\to\infty$ does not occur for any $n$, as $k\to\infty$. What is known about this aspect of the problem?

Answer 1

I should have looked harder: a 2010 survey of Jeff Lagarias on p.22 conjecture (C2) exactly answers this (as of 2010, of course):

Does the 3x + 1 function have a divergent trajectory, i.e., an integer starting value whose iterates are un- bounded? This is conjectured not to be the case.