The Collatz Conjecture is well known with the sequence
$$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$; the sequence converging $1$ (so called oneness).
Is there any conjecture/theorem on whether the sequence would converge for any other value of $k$; or could it be shown that the sequence diverges for values of $k$ other than $3$?
By convergence here I mean that the sequence after finite steps ends with a stable fixed number such as in case of Collatz it is the case with the number $1$.
Append: In the mean time I wrote out a conjecture on this over here >>>, for those who might be interested.
It can be shown that for $k=5$ the sequence can "converge" to other numbers rather than $1$:
$$13\to 66\to 33\to 166 \to 83 \to 416\to 208\to 104\to 52\to 26\to 13.$$