Berarducci–Peterzil–Pillay's interpretability conjecture for finite central extensions

Let $G$ be a definably connected definable group in an o-minimal structure $M$, and let $Z$ be a finite central subgroup. Write $\Ext_d(G,Z)$ for the definable second cohomology group and $\Ext(G,Z)$ for the abstract second cohomology group. The natural inclusion

$$\Ext_d(G,Z)\hookrightarrow \Ext(G,Z)$$

is defined by viewing definable extensions as abstract extensions.

Berarducci–Peterzil–Pillay's conjecture. The natural inclusion

$$\Ext_d(G,Z)\hookrightarrow \Ext(G,Z)$$

is an isomorphism.

This is the cohomological reformulation of the conjecture that every abstract finite central extension of a definably connected definable group is naturally interpretable in the o-minimal structure. It is known for abelian groups and, by the theorem stated in the paper, for definably connected solvable definable groups; the general case remains open.