The integer-complexity conjecture for powers of 2 and 3

Let $\|n\|$ denote the integer complexity of a positive integer $n$, namely the minimal number of $1$'s needed to express $n$ using addition, multiplication, and parentheses.

Integer-complexity conjecture. For $a\geq 1$ and $b\geq 0$,

$$\|2^a\cdot 3^b\|=2a+3b.$$

This conjecture incorporates Selfridge's question of whether some power of $2$ has complexity less than $2a$. It asserts that products of powers of $2$ and $3$ attain the expected complexity, but the supplied text does not indicate whether the conjecture has been resolved.