The integer complexity conjecture for 2-, 3-, and 5-smooth numbers with bounded 5-adic exponent

Let $a,b,c$ be nonnegative integers satisfying $a+b+c>0$ and $c<6$. For a positive integer $n$, let $\|n\|$ denote its integer complexity, namely the least number of ones needed to build $n$ using addition and multiplication. Integer complexity conjecture.

$$\|2^a3^b5^c\|=2a+3b+5c.$$

This conjecture generalizes Selfridge's question in the case $c=0$. The stated computational verification below $10^{12}$ supports it, but the conjecture remains open; if true, it would support the lower bound $\limsup_{n\to\infty}\|n\|/\log n\geq 2/\log 2$.