Canonical splitting conjecture for the regulator-value space

Let $N\geq 3$, and let ${\mathcal L}_{-1}$ and ${\mathcal L}_0$ be the ${\mathbb Q}$-subspaces of ${\mathbb R}$ generated by the images of the regulator pairings in the first and third columns of the relative-cohomology diagram. Let ${\mathcal L}$ be the image of the regulator pairing in the middle column.

Canonical splitting conjecture. There is a canonical direct-sum decomposition

$${\mathcal L}={\mathcal L}_{-1}\oplus {\mathcal L}_0.$$

The paper notes that ${\mathcal L}_{-1}+{\mathcal L}_0\subseteq {\mathcal L}$ and presents the canonical splitting as a stronger expected property beyond mixed linear independence.