The powers-of-three complexity conjecture

Let $O=\{1,S,+,*\}$ and $O_{|}=\{1,S,+,*\}$ denote the two operation sets considered, and let $c_O(n)$ and $c_{O_{|}}(n)$ be their corresponding complexities.

Powers-of-three complexity conjecture. For every $k\in\mathbb N$,

$$c_O(3^k)=4k-1.$$

Alternatively, for every $k\in\mathbb N$,

$$c_{O_{|}}(2^k)=2k.$$

The source immediately notes that the lower-bound result implies this conjecture cannot hold simultaneously with the asymptotic complexity conjecture; accordingly, this candidate is refuted.