Nature of the Cooper-pair insulator beyond the transition
In plain words
On the insulating side, some films such as amorphous indium oxide and titanium nitride show a giant resistance peak in a magnetic field and signs that electrons are still bound in pairs. How paired electrons form an insulator, and what sets its conduction, is open.
Precise statement
In amorphous InOx, TiN and nanopatterned Bi films on the insulating side of the field-tuned transition, $\rho \sim \operatorname{exp}(T_{0}/T)$ with $T_{0}$ peaking at a field $B_{\mathrm{peak}}$, the magnetoresistance peak spans several orders of magnitude, and patterned films show $h/2e$ periodic magnetoresistance (Stewart et al. 2007). Explain the pair-localization mechanism and $T_{0}(B)$, and determine whether abrupt current jumps in the $I-V$ curves reflect electron overheating (Altshuler, Kravtsov, Lerner and Aleiner 2009) or a distinct phase (proposed superinsulator).
What would settle it
Electron-temperature measurement during nonlinear transport plus local probes of pair density in the insulator, compared with a theory of localized preformed pairs.