Logarithmic delay and polynomial maximum-excursion bounds for Collatz sequences

Let $d_{\text{T}}(n)$ denote the delay and $M_{\text{T}}(n)$ the maximum excursion of the Collatz sequence starting at the positive integer $n$. Delay and maximum-excursion bound conjecture. There exist positive reals $\alpha$ and $\beta$ such that, for every integer $n\geq2$,

$$d_{\text{T}}(n)\leq\alpha\,\log n$$

and

$$M_{\text{T}}(n)\leq n^\beta.$$

These weak bounds are proposed as heuristic formulations sufficient for the paper's purpose; no resolution is supplied.