Polynomial-growth conjecture for the least ternary height representative

For a natural number $h$, let $n_h$ be the smallest ternary integer, if it exists, such that $A(n_h)=h$. Polynomial-growth conjecture for the least ternary height representative. There exist constants $E_1$ and $E_2$ such that

$$h^{E_1}\ll n_h\ll h^{E_2},\qquad E_1\ge3.$$

The known bound $A(pqr)\le p-1$ gives the lower-growth constraint $E_1\ge3$; the conjectured polynomial upper bound is not established in the supplied text.