Linear independence of even Dirichlet beta values

Let $\beta(s)=1-3^{-s}+5^{-s}-\cdots$ denote the Dirichlet beta function. For all sufficiently large even integers $s$, we prove $\dim_{\mathbb{Q}}\mathrm{span}_{\mathbb{Q}}\{\beta(2),\beta(4),\ldots,\beta(s)\}\geq 0.21\sqrt{s/\log s}$.