Iterated log-sine basis conjecture at pi over 3

Let $\mathcal{L}^{o}_{k}$ and $\mathcal{L}^{e}_{k}$ denote the spaces of iterated log-sine integrals at $\pi/3$ of weight $k$ with odd and even $d=|\mathbf{k}-\mathbf{1}_{n}-\mathbf{l}|$, respectively, and let $S^{o}_{k,k}$ and $S^{e}_{k,k}$ be the corresponding shifted log-sine sets. Iterated log-sine basis conjecture. The set of real numbers $S^{o}_{k,k}$ is a basis of $\mathcal{L}^{o}_{k}$, $S^{e}_{k,k}$ is a basis of $\mathcal{L}^{e}_{k}$, and $S^{o}_{k,k}\cup S^{e}_{k,k}$ is a basis of $\mathcal{L}^{o}_{k}+\mathcal{L}^{e}_{k}$. The source gives only dimension upper bounds and no proof of these basis assertions, so they remain open.