M. Stein's dimension conjecture for linear relations among recursive-sequence Dirichlet series

Let $\Phi_{2s}$, $\Phi_{2s}^{*}$, $\Psi_{2s}$, and $\Psi_{2s}^{*}$ be the Dirichlet series defined in the paper, and let $V_m$ be the set of all $\left(t_1,\ldots,t_{4m}\right)\in\mathbb{Q}^{4m}$ satisfying

$$\sum_{s=1}^{m}\left(t_{4s-3}\Phi_{2s}+t_{4s-2}\Phi_{2s}^{*}+t_{4s-1}\Psi_{2s}+t_{4s}\Psi_{2s}^{*}\right)=0.$$

M. Stein's conjecture. One has $\dim_{\mathbb{Q}}V_m=m$ for every positive integer $m$. Moreover, for every $\left(t_1,\ldots,t_{4m}\right)\in V_m$,

$$t_{4s}=0\quad\text{for }2\mid s,\qquad\text{and}\qquad t_{4s-1}=0\quad\text{for }2\nmid s.$$

The claim describes the dimension and the parity-dependent vanishing pattern of all rational linear relations among the indicated Dirichlet series. It is presented in the source as a conjecture attributed to M. Stein; the supplied material gives no evidence that it has been proved or disproved.