The logarithmic lower-bound conjecture for the Schemmel iteration function $R_2$

Let $R_2$ be the function introduced in the paper, and let $x$ be an odd integer greater than $3$. The logarithmic lower-bound conjecture.

$$R_2(x)\geq \frac{\log \left(\frac{49}{15}x\right)}{\log 7}.$$

The conjecture proposes a lower bound for the number of iterations measured by $R_2$. The paper gives numerical motivation and discusses upper bounds, but does not establish this lower bound.