Weak mixed linear independence conjecture for regulator-value spaces

Let $N\geq 3$ be an integer. Let ${\mathcal L}_{-1}$ be the regulator-value subspace in weight $2$, and let ${\mathcal L}_0^\sharp$ be the regulator-value subspace obtained from the sharp part of the weight-$1$ motivic cohomology; explicitly, the latter is generated by logarithms of sums of primitive $N$-th roots of unity.

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

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

i.e. ${\mathcal L}_{-1}$ and ${\mathcal L}_0^\sharp$ are linearly independent over ${\mathbb Q}$. This is the weaker hypothesis used to obtain the sharp-part result when the full mixed independence conjecture is unavailable.