The 3x+1 maximum excursion constant conjecture

For the $3x+1$ map, let $t(n;\omega)$ be the maximum-excursion statistic in the repeated random walk model, and define

$$\rho:=\limsup_{n\to\infty}\frac{\log t(n;\omega)}{\log n}.$$

3x+1 maximum excursion constant conjecture. The maximum excursion constant $\rho$ is finite and

$$\rho=2.$$

The conjecture is motivated by the corresponding repeated random walk model theorem and its density estimates. The source does not provide a proof for the actual $3x+1$ map.