My tentative answer is that the conducting sphere falls faster, but there may be another subtle effect that I'm missing.

The important difference between the two spheres (conducting and insulating), is that the conducting sphere must maintain a constant potential across its surface, which essentially requires the sheath to be the same width at the top and bottom. In contrast, the movement of the insulating sphere means the sheath is larger on the top than the bottom, essentially creating a polarization drag as it falls.

In a little more detail, let's work out the size of the sheath for the insulating sphere. Neither sphere draws any net current so we can calculate this by equating the ion and electron currents. The ion current is simply given by the Bohm criterion -- ions move into the object at the Bohm velocity -- giving \[j_i=en_e (c_s\pm v_s).\]Here \(c_s =\sqrt{k_B T_e/m_i}\) is the ion sound speed (Bohm velocity), \(e\) is the electron charge, \(n_e\) and \(T_e\) are the electron density and temperature, \(m_i\) is the ion mass and \(v_s\) is the speed of the sphere. The + accounts for the extra ion flux on the bottom of the sphere, while the - accounts for decreased flux on the top (as it moves downwards).

The electron current is given by assuming a Maxwellian distribution function and integrating up to the potential of the sphere, \(\Delta V\). This gives \[j_e = \sqrt{\frac{k_B T_e}{2\pi m_e}}e^{-\frac{e\Delta V}{k_B T_e}}.\]We then equate ion and electron currents to solve for \(\Delta V\), giving\[\Delta V=\frac{k_B T_e}{2e} \ln \left( \frac{m_i}{2\pi m_e} \left(1\mp \frac{2v_s}{c_s} \right)\right), \]where the - is for the bottom and the + is for the top. Thus, we see that there is a larger potential difference between the plasma and the top of the sphere than the plasma and the bottom of the sphere. The electric field associated with this is pointing towards the sphere, which is negatively charged, meaning there is a larger upwards force on the top than downwards force on the bottom -- i.e., a drag.

For the conducting sphere, charges will move around on the surface so as to exactly keep the potential constant, counteracting this effect. Thus it falls as if in a neutral gas.