Antihydra's odd-even frequency conjecture

Consider the map $H:^+\to^+$ defined by

$$H(x)=\begin{cases}3\frac{x}{2}&\text{if }x\text{ is even},\\3\frac{x-1}{2}&\text{if }x\text{ is odd}. \end{cases}$$

Starting from $x=8$, classify each iterate as odd or even. Antihydra's odd-even frequency conjecture. At no point in the iteration are there strictly more than twice as many odd numbers as even numbers. This conjecture is presented as equivalent to nonhalting of the six-state Antihydra Turing machine from the all-zero tape. The source does not state whether the conjecture itself has been proved or disproved.