The divergence conjecture for the shortened qx+1 map

Let $q$ be an odd prime, and let $\chi_q:\mathbb{Z}_2\to\mathbb{Z}$ denote the numen of the shortened $qx+1$ map. Divergence conjecture for $qx+1$. There exists a $\mathfrak z\in\mathbb{Z}_2\setminus\mathbb{Q}$ such that $\chi_q(\mathfrak z)\in\mathbb{Z}$ if and only if $q=3$. The claim links integral values of the numen at irrational 2-adic inputs to the classical $3x+1$ case and is unresolved in the source.