  This article is concerned with stochastic differential equations driven by a
$d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$,
understood in the rough paths sense. Whenever the coefficients of the equation
satisfy a uniform hypoellipticity condition, we establish a sharp local
estimate on the associated control distance function and a sharp local lower
estimate on the density of the solution. Our methodology relies heavily on the
rough paths structure of the equation.
