Suppose we repeatedly apply the transformation $$a_{k+1}= \begin{cases}\frac{a_k}{q} & \text{when } a_k \equiv 0 \pmod{q} \text{; and}\\ \lceil \frac{pa_k}{q} \rceil & \text{otherwise.} \end{cases} $$ Where $a_0$ is a natural number and $p$ and $q$ are distinct primes.
Are there choices of $a_0$, $p$, and $q$ such that the resulting sequence diverges indefinitely, and will not ever loop through the same numbers?
Of course this is just a more general form of the $3n+1$ (or, Collatz) conjecture.
I'm aware of the statistical argument for some choices of $p$ and $q$ diverging; and in fact even with small numbers like $a_0=7$, $p=5$, and $q=2$ the sequence seems to diverge. But might it be the case that for any choice of $a_0$, $p$, and $q$ we will always eventually encounter a number divisible by such a great power of $q$ that our sequence converges into a loop?