The generalized Kaprekar height conjecture

Fix a base $b\geq 2$. Write every $u\geq 1$ uniquely as a tower of powers of $b$ ending in $0<\omega\leq 1$, and let $\operatorname{ht}(u)$ be the height of this tower. Let $K(n)$ denote the smallest number with exactly $n$ generators in base $b$. Generalized Kaprekar height conjecture.

(i) If $b=2$ and $n\geq 2$, then

$$\operatorname{ht}(K(n))=\left\lceil\log_2(n)\right\rceil+3.$$

(ii) If $b=3$ and $n\geq 3$, then

$$\operatorname{ht}(K(n))=\left\lceil\log_2(n/5)\right\rceil+4.$$

(iii) If $b\geq 4$ is even and $n\geq 2$, then

$$\operatorname{ht}(K(n))=\left\lceil\log_2(n)\right\rceil+2.$$

(iv) If $b\geq 5$ is odd and $n\geq 2$, then

$$\operatorname{ht}(K(n))=\left\lceil\log_2(n)\right\rceil+1.$$

The conjecture is motivated by computed tables and describes the rapid growth of $K(n)$ through exponential-tower height. The source offers computational evidence but no proof of these formulas.