Conjecture on linear independence of twisted periods with different indices

Let $K$ be the weight, $S_K$ the space of cusp forms under consideration, and $r_{\ell,\chi}$ the twisted period indexed by $\ell$ and a primitive Dirichlet character $\chi$. Let $3\leq\ell_1<\ell_2<\cdots<\ell_n\leq\frac{K-4}{2}$ be integers, let $D\geq1$ be an integer, and let $\chi$ be a primitive Dirichlet character satisfying $\chi(-1)=(-1)^{\ell_i}$ for every $i$. Different-index twisted-period conjecture. If $n\leq\dim S_K$, then $\{r_{\ell_i,\chi}\}_{i=1}^{n}$ is linearly independent. The conjecture predicts that twisted periods with one fixed character and different indices span the relevant dual space; it was verified in the paper for $K,D\leq40$ by checking nonsingularity of the associated coefficient matrices.