The 1439 extremal conjecture for integer complexity

For each integer $n>1$, let $\|n\|$ denote its integer complexity, the least number of ones needed to construct $n$ using addition and multiplication. Then

The 1439 extremal conjecture. For all $n>1$, one has

$$\frac{\|n\|}{\log n} \leq \frac{26}{\log 1439},$$

with equality obtained only at $n=1439$.

The statement is presented as the obvious conjecture for primes other than $2$ or $3$, but the source explicitly notes that it is false; the conjecture is therefore refuted.