The efficient-prime finite-control conjecture for integer complexity

For real $\alpha$ and integers $m<n$, define $S(m,n,\alpha)$ to mean that if

$$\|k\|\leq \alpha\log k$$

for every integer $k$ with $m<k<n$, then

$$\|n\|\leq \alpha\log n.$$

For a prime $p$, let $v_p$ denote the $p$-adic valuation. The conjecture concerns the $p$-adic behavior of efficient primes.

The efficient-prime finite-control conjecture. For any $\alpha>\frac{2}{\log 2}$, there is a finite set of primes $S_\alpha$ such that, for any $m$, if $v_p$ is sufficiently large for all $p\in S_\alpha$, then $S(m,n,\alpha)$.

The source says that a careful study of efficient primes is expected to prove this assertion, but gives no resolution. The quantification and the role of $n$ in the final condition follow the supplied statement, whose formulation may warrant checking.