Handelman's integrability conjecture for log-convex functions

Let $h:(0,\infty)\rightarrow(0,\infty)$ be a log-convex function satisfying

$$\lim_{n\rightarrow\infty}\frac{h(n)}{h(n+1)}=1.$$

Assume that $h^2(x)/h(2x)$ is integrable on $(0,\infty)$. Handelman's integrability conjecture. Then $h$ is also integrable on $(0,\infty)$. Handelman posed this question in connection with maximal functions and the integrability of products involving log-convex functions; the supplied text does not establish whether the conjecture is resolved.