Bounded-height record-setter asymptotic conjecture for the function z

Let $C>1$ be fixed, and let $n_1<n_2<\cdots$ be an increasing sequence of positive integers such that $h(n_i)\leq C$ for all $i$ and

$$\lim_{i\rightarrow\infty}\tau(n_i)=\infty.$$

Bounded-height asymptotic conjecture. Then

$$z(n_i)=o(\log\tau(n_i)).$$

This concerns the growth of the arithmetic function $z$ along sequences whose associated height $h$ remains bounded while the divisor-counting function $\tau$ tends to infinity. The supplied text gives no resolution status.