Relations among rational coefficients in reciprocal hyperbolic series

For each positive integer $p$, let $c_p,d_p,e_p,f_p,g_p,h_p,i_p,j_p,k_p,l_p$ be the rational coefficients defined by the evaluations of the reciprocal hyperbolic series at $\pi$. Coefficient relations. The following identities hold:

$$c_p=\displaystyle\!\frac{1}{2^{4p-2}}g_p,\qquad d_p=-\displaystyle\!\frac{1}{2^{4p-2}}h_p=\displaystyle\!\frac{(-1)^p}{2^{6p-3}}k_p,\\ e_p=-\displaystyle\!\frac{1}{2^{4p}}i_p=\displaystyle\!\frac{(-1)^p}{2^{6p}}l_p,\qquad f_p=\displaystyle\!\frac{1}{2^{4p}}j_p.$$

These identities express conjectural relations among the rational coefficients arising in the gamma-function evaluations of the series. They are presented on the basis of computed examples, and no proof or resolution is supplied in the source.