Cover time when the connectivity-threshold correction is positive

Determine the behavior of the cover time of the giant component of a d-dimensional random geometric graph when r^d=(log n+f(n))/V_d, |f(n)|=o(log n), and f(n)>0.

Background

The connectivity threshold is studied in a window described by r^d=(log n+f(n))/V_d, where f(n) may be positive or negative and is sublogarithmic in magnitude. The paper obtains a degree-one-vertex lower bound when f(n) is negative, but does not determine the corresponding cover-time behavior on the positive side of the threshold. This question concerns the unresolved transition behavior within that threshold window.