Erdős [Er74b] proved that $\limsup h(x)/x= \infty$, and claimed a similar proof for this problem. A complete proof that $h(x)/x\to \infty$ was provided by Pollack and Pomerance [PoPo16].

A similar question can be asked if we replace the condition $(a,b)=1$ with the condition that $a$ and $b$ are squarefree.