The even-iterate frequency conjecture for the Collatz map

Let $T:\mathbb N\to\mathbb N$ be the Collatz function, let $T^{(k)}$ denote its $k$-fold iterate, and fix $n_0\in\mathbb N$ with $n_0\geq 1$. The even-iterate frequency conjecture. The proportion of even numbers among the sequence of iterates $T^{(k)}(n_0)$ tends to $1/2$ as $k$ tends to infinity. This is presented as an equivalent version of the Collatz conjecture in the paper and is therefore open with the Collatz problem.