The structural decomposition conjecture for definable groups in dense pairs

Let $\langle G,*\rangle$ be a definable group. A group is small if it is a definable set of the corresponding smallness notion in the dense expansion, and a group is $\bigvee$-definable if it is a directed union of definable sets. Then there is a short exact sequence

$$0\longrightarrow \mathcal B\longrightarrow \mathcal U\xrightarrow{\tau}K\longrightarrow 0$$

with quotient map to $G$ represented by the diagram in the conjecture, such that $\mathcal U$ is $\bigvee$-definable, $\mathcal B$ is $\bigvee$-definable in $\mathcal L$ and satisfies

$$\dim(B)=\operatorname{ldim}(G),$$

$K$ is definable and small, $\tau:\mathcal U\to G$ is a surjective group homomorphism, and all maps involved are $\bigvee$-definable. Structural decomposition conjecture. Every definable group admits such a short exact sequence, decomposing it into an $\mathcal L$-definable component of full $\operatorname{ldim}$ and a definable small component. The conjecture seeks a general structural description of definable groups in expansions by a dense set; the paper presents it as an ultimate goal motivated by analyses of semi-bounded groups. Its resolution is not established in the supplied text.