Erdős Problem #692 — Unimodality of Divisor Densities

For natural numbers $a,b$, let

$$E(a,b)=\{x\in\mathbb N:|\{d\in\mathbb N:a<d<b\text{ and }d\mid x\}|=1\}.$$

Let $\delta_1(a,b)$ be the density of $E(a,b)$. Is it true that, for every function $\delta:\mathbb N\to\mathbb N\to\mathbb R$ such that $\delta(a,b)$ is the density of $E(a,b)$ for every $a,b\in\mathbb N$, and for every $n\in\mathbb N$, the function $m\mapsto\delta(n,m)$ is unimodular on the set of natural numbers $m\geq n+1$—that is, it increases up to some point and decreases thereafter? The assertion is false.