Conjecture on linear independence of twisted periods with different characters

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 $D\geq1$ and $3\leq\ell\leq\frac{K-4}{2}$ be integers, and let $\chi_1,\ldots,\chi_n$ be distinct primitive Dirichlet characters modulo $D$ satisfying $\chi_i(-1)=(-1)^\ell$. Different-character twisted-period conjecture. If $n\leq\dim S_K$, then the set $\{r_{\ell,\chi_i}\}_{i=1}^{n}$ of twisted periods on $S_K$ is linearly independent. The conjecture predicts that fixed-index twisted periods with different characters span $S_K^*$ when sufficiently many primitive characters are available; it was verified in the paper for $K,D\leq40$ by the corresponding matrix computations.