Arithmeticity conjecture for fifth-moment Berndt-type integrals

Let $\Gamma=\Gamma(1/4)$ and, for $n\geq1$, define

$$J_{2n-1}=\displaystyle\!\int_0^\infty\displaystyle\!\frac{x^5\,dx}{(\cos x+\cosh x)^{2n-1}},\qquad J_{2n}=\displaystyle\!\int_0^\infty\displaystyle\!\frac{x^5\,dx}{(\cos x+\cosh x)^{2n}}.$$

Here $\mathbb{Q}$ denotes the rational numbers. Fifth-moment arithmeticity conjecture. For every integer $n\geq1$,

$$J_{2n-1}\in\displaystyle\!\frac{\Gamma^4}{\pi}\mathbb{Q}+\displaystyle\!\frac{\Gamma^4}{\pi^2}\mathbb{Q}+\displaystyle\!\sum_{j=2}^{n-3}\displaystyle\!\sum_{i=1}^{6}\displaystyle\!\frac{\Gamma^{8j-4}}{\pi^{6j-10+i}}\mathbb{Q}+\displaystyle\!\sum_{j=1}^{3}\displaystyle\!\sum_{i=1}^{7-2j}\displaystyle\!\frac{\Gamma^{8j+8n-20}}{\pi^{6j+6n-22+i}}\mathbb{Q},$$

and

$$J_{2n}\in\mathbb{Q}+\displaystyle\!\sum_{i=1}^{4}\displaystyle\!\frac{\Gamma^8}{\pi^{i+1}}\mathbb{Q}+\displaystyle\!\sum_{\substack{i=1\i\ne2}}^{6}\displaystyle\!\frac{\Gamma^{16}}{\pi^{i+5}}\mathbb{Q}+\displaystyle\!\sum_{j=3}^{n-2}\displaystyle\!\sum_{i=1}^{6}\displaystyle\!\frac{\Gamma^{8j}}{\pi^{6j-7+i}}\mathbb{Q}+\displaystyle\!\sum_{j=1}^{3}\displaystyle\!\sum_{i=1}^{7-2j}\displaystyle\!\frac{\Gamma^{8j+8n-16}}{\pi^{6j+6n-19+i}}\mathbb{Q}.$$

These membership assertions refine the paper's proven degree bounds for related integrals and are supported by extensive numerical evidence, but remain conjectural.