Conjecture on the minimal orders of ordered and additive factorizations

Let $g(n)$ count ordered factorizations of $n$, let $b(n)$ count the corresponding additive factorizations, and let $a(n)$ be the associated counting function. Let $\rho$ be the constant used in the paper. For positive quantities $f$ and $h$, the notation $f(n)\asymp h(n)$ means that their ratio is bounded above and below by positive constants.

Minimal-order conjecture. As $n\to\infty$,

$$b(n)\asymp n^{\rho-1}\quad\text{for composite }n,$$

and

$$a(n)\asymp n^{\rho-1}\quad\text{for all }n.$$

This concerns the poorly understood minimal orders of the functions $a(n)$ and $b(n)$; the preceding discussion notes that the simple prime values of $b(n)$ do not determine the minimal order of $a(n)$, so these asymptotics remain open.