Coefficient relations for Berndt-type integrals

For every positive integer $p$, let $i_p$, $j_p$, $g_p$, and $h_p$ be the numbers appearing in Theorems and. Coefficient relations. The following identities should hold:

$$g_p=-\displaystyle\!\frac{2^{2p-1}-(-1)^p}{2^{2p-1}}i_p, \qquad h_p=-\displaystyle\!\frac{2^{2p-1}+(-1)^p}{2^{2p-1}}j_p.$$

The conjecture is motivated by comparing the coefficients in the paper's examples for the associated integrals. Its resolution is not specified in the source.