The exact integer complexity conjecture for powers of 2 and 3

Let $\|n\|$ denote the least number of ones needed to write the positive integer $n$ using addition, multiplication, and parentheses. For integers $a,k\geq 0$ with $a+k\geq 1$, consider the integer $2^a\cdot 3^k$. Exact integer complexity conjecture. For all $a\geq 0$ and all $k\geq 0$ with $a+k\geq 1$, one has

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

This asserts that the standard factorization gives an optimal expression for every positive integer composed only of the primes $2$ and $3$. The statement was supported by the exact results for several ranges of $a$ and by computational verification up to $10^{12}$, but its general status is open.