The 2-complexity conjecture for powers of 6 and 10

Let $\|n\|_2$ denote the $2$-complexity of an even positive integer $n$, namely the minimal number of $2$'s needed to express $n$ using addition, multiplication, and parentheses.

2-complexity conjecture. For $m\geq0$ and $r\geq1$,

$$\|2^m\cdot 6^r\|_2=m+3r$$

and

$$\|2^m\cdot 10^r\|_2=m+4r.$$

The conjecture extends the preceding theorem, which proves these equalities only for bounded values of $r$ and gives partial bounds in two additional cases. The paper explicitly cautions that the claim may fail for large $r$, and the supplied text gives no resolution.