This is a continuation of my former question: On the class number

It appears that the problem of showing that every positive integer is the class number of some quadratic number field seems rather intractable, despite the wide belief that it is true.

So I ask an easier question: For each positive number $x$, let $\rho(x)$ denote the number of positive integers $n \leq x$ such that $n$ cannot be realized as the class number of a quadratic number field. Then is it possible to show that $\rho(x)/x$ tends to 0? Better yet, is it possible to show that $\rho(x) \ll x^\delta$ for some $0 < \delta < 1$?

If $\rho(x) \ll x^\delta$ isn't a 'natural' upperbound, then what would be? Can one expect a better or worse upper bound? Can a bound of the form $\rho(x) \ll x/(\log x)^A$ for some $A > 0$ be likely?

Thanks for any information.