  We establish Harnack inequalities for stochastic differential equations
(SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter
$H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift
which satisfies a one-sided Lipschitz condition, otherwise we still get
Harnack-type estimates, but the constants will, in general, depend on the space
dimension. Our proof is based on a coupling argument and a regularization
argument for the time-change.
