Gordon–Pomerance conjecture for integers of prescribed rank of apparition

For $z\in\mathbb N$, let $\mathcal B_z=\{n\in\mathbb N:z(n)=z\}$, where $z(n)$ denotes the relevant rank-of-apparition function, and write $L(t)$ for the paper's slowly varying logarithmic function. For sufficiently large $t$, uniformly in $z$,

$$\#\mathcal B_z(t)\leq\frac{t}{L(t)^{1+o(1)}}.$$

Gordon–Pomerance conjecture. With the same hypotheses as Lemma Gordon–Pomerance, the displayed upper bound holds. The conjecture is used to obtain the conditional upper bound for the moment sums; the source gives heuristic, analogical, and empirical support but no proof.