The usual argument in favor of the Collatz conjecture (or at least in favor of there being no unbounded trajectories) essentially argues that, if we have the "shortcut" function defined by: $$f(2x)=x$$ $$f(2x+1)=3x+2$$ then the parity of a number in the trajectory $x,\,f(x),\,f^2(x),\ldots$ of iterates of the Collatz function should be randomly distributed and therefore that the geometric mean of the ratios $\frac{f^n(x)}{f^{n+1}(x)}$ should be $\sqrt{3/4}$ unless $x$ is small - which suggests that the sequence should not grow without bound.
This is famously an open question - but there's an obvious generalization of the same reasoning: fix some natural number $m$ and some sequence of integers $a_k$ and $b_k$ indexed over the set $\{0,\ldots,m-1\}$. Define a function by the rule: $$g(mx+k)=a_kx+b_k$$ where $x$ is an integer and $k$ is an integer in $[0,m)$. Let's say that $g$ is Collatz-like if:
$\gcd(m,a_k)=1$ for every $k$.
$\prod_{k=0}^{m-1}\frac{a_k}m$ < 1.
There is some $k$ such that $\frac{a_k}m > 1$.
The first condition enforces that the proportion of integers $x$ with a prescribed sequence of mod $m$ residues for $x,\,f(x),\,f^2(x),\ldots,\,f^{\ell}(x)$ is exactly $1/m^{\ell}$, which justifies treating these residues somewhat randomly. The second enforces that the logs of large numbers are expected to decrease. The last is a non-triviality condition, since any function violating it clearly has no unbounded trajectories.
Is there any example of a Collatz-like function $g$ for which it is known whether $g$ has an unbounded trajectory?
