Mixed linear independence conjecture for partial Dirichlet $L$-values

Let $N\geq 3$ be an integer. 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 weights $2$ and $1$, respectively. The first is generated by the values $L^{(N)'}(-1,x)$, and the second by the corresponding values $L^{(N)'}(0,x)$, together with $\log|\zeta_N+\overline{\zeta_N}|$ when $N=4p^r$.

Mixed linear independence conjecture. As ${\mathbb Q}$-subspaces of ${\mathbb R}$,

$${\mathcal L}_{-1}\cap {\mathcal L}_0=\{0\},$$

i.e. ${\mathcal L}_{-1}$ and ${\mathcal L}_0$ are linearly independent over ${\mathbb Q}$. The paper introduces this because the mixed linear independence is not known, although it appears generically to hold, and uses it to study the relative motivic cohomology as a $G$-module.