Linear independence of shifted log-sine integrals

Let $\operatorname{SLs}(k_{1},\dots,k_{n})$ denote a shifted log-sine integral, with $m,n\geq 0$ and $k_i\geq 2$. Linear-independence conjecture. The elements of

$$\left\{\pi^{m}\operatorname{SLs}(k_{1},\dots,k_{n}) \mathrel{}\middle|\mathrel{} m \ge 0, n \ge 0, k_{i} \ge 2\right\}$$

are $\mathbb{Q}$-linearly independent. Numerical computations found no $\mathbb{Q}$-linear relations for values satisfying $m+k_{1}+\dots+k_{n}\leq 8$, but the general assertion remains open.