= Solution
Assume both densities are nonnegative and $\Omega_{{\rm DE},0}<1$, so a nonzero <pressureless matter> component exists. Their ratio obeys
$$
\boxed{\frac{\rho_{\rm DE}}{\rho_M}=\frac{\Omega_{{\rm DE},0}}{1-\Omega_{{\rm DE},0}}a^{-3w}\longrightarrow0\quad(a\to0,\ w<0).}
$$
Thus <matter domination> gives
$$
\boxed{H(a)\sim H_0\sqrt{1-\Omega_{{\rm DE},0}}\,a^{-3/2}.}
$$
At fixed $H_0$, the early expansion rate does depend on the present <dark energy> fraction: increasing it reduces $\rho_{M,0}=(1-\Omega_{{\rm DE},0})3H_0^2/(8\pi G)$. There is no contradiction with negligible early <dark energy>, since the leading expression is simply $H^2\sim8\pi G\rho_M/3$. At fixed physical matter density, that leading rate is independent of the negligible component.
The endpoint $\Omega_{{\rm DE},0}=1$ has no matter, so the ratio argument does not apply. Its <Hubble parameter> is $H=H_0a^{-3(1+w)/2}$: it diverges for $-1<w<0$, is constant for $w=-1$, and tends to zero for $w<-1$.
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