The Collatz chain descent conjecture

Let $C_n(L)$ denote the Collatz chain associated with an odd integer $L$ in the interval under discussion, and let $\alpha(C_n)$ and $\beta(C_n)$ denote the numbers of applications of the odd and even steps, respectively. Assume that odd integers in $[2^n+1,2^n]$ are very close to $2^{n+1}$. If

$$2^{\beta(C_n)}>2\cdot 3^{\alpha(C_n)}$$

so that

$$2^{\beta(C_n)-1}>3^{\alpha(C_n)}$$

and

$$\frac{2^{\beta(C_n)-1}}{3^{\alpha(C_n)}}>1,$$

then either $C_n(L)$ lies in $[1,2^n]$, or it starts with a chain that lies in $[1,2^n]$. Equivalently,

$$C_n(L)-\frac{3^{\alpha(C_n)}L}{2^{\beta(C_n)}}\leq 2^n-\frac{3^{\alpha(C_n)}L}{2^{\beta(C_n)}}$$

or it starts with such a chain.